There's definitely nothing in the official books, 3.5 or 4e. Here's a guesstimate.
Anti-sodomy laws (anti-lesbian laws are an adjucant to these statutes) are based mostly on the presumption that reproduction is the obligation of the members of a society. For instance, the famous Leviticus ban comes during a time when the Israelites were losing men by the thousands in regional wars, and they needed more bodies.
So would a similar law be in place on Eberron? The only nations this might make sense for are Karrnath and Darguun. While the Karrnathi recently started using necromancy on a military scale, their martial tradition (and warlord regional rule) goes back over a thousand years, so a surplus of young men would be a must. Their law may criminalize homosexuality, but to what extent? Fines? Exile? Execution? And Lhesh Haruuc needs as many goblins as he can get his hands on; though, goblin law is less rigorous, so it may be that they just lynch gay goblins.
Elsewhere, I strongly doubt it. Any place with elven culture (Valenar, the racially diverse Sharn and Breland) would have to allow for their heterodox gender roles. Elves, living as long as they do, have a strong influence on any laws concerning their personal lives. Rural areas like the Reaches and the Marches probably mirror the "Two-Spirit" tradition, allowing same-sex couples as an accommodation. Pirate societies were known to accept homosexual unions, so the Principalities probably even let ship captains wed same-sex couples.
There's no way the Sisters Three are wasting time on worrying about LBGT monsters. The Mror Holds are set up in a clan system, and tribal systems either accept or reject homosexuality, and sometimes only certain types; that's a hard one to guess. Aundair, at least on the rich side, is an intellectual society, so it's difficult to find a philosophical foundation for any anti-gay statutes, and Aurala doesn't need more bodies, she needs better magic, so I doubt she's concerned with homosexuality.
Thrane is a maybe, because it falls into the traditional Judeo-Christian idea of a single deity, which is common for plains dwellers. Their military strategy is definitely to pour troops onto a problem, so they'd need more able-bodied soldiers. But nothing in the official texts paints the Silver Flame as explicitly heteronormative.
To my mind it was very looked down on when Galifar was united, and prohibited by law, but when Galifar was divided the different nations evolved their own view of it over time.
Aundair is often the butt of many jokes about how effeminate and homosexual their men are, but doesn't actually have a higher incidence of homosexuality among its population. This role used to be attributed more to Cyre, but Aundair was always a favored target. Because of that perception, the lower classes of Aundair are prone to defending their heterosexuality, and therefore attacking homosexuality. So most lower class gay men and women keep their sexuality to themselves. Its more common, and somewhat more open (in closed circles) among the nobility. They don't go public very often, but among themselves they are accepting (generally) of it in a purely sexual sense. Actual love for another person of the same sex is considered a mental defect.
I'm going to say that Thrane is very tolerant of homosexuality, because the Church is too often portrayed as bigoted bullies. The work of the Keepers over centuries has greatly weakened the old prejudices against homosexuality, and while their is still a lingering feeling of it being "Gross", and most of the citizenry would still be rather distant with open homosexuals, it is not openly persecuted or considered morally wrong. Just different, which is enough to put many social barriers on homosexual men and women.
Karrnath's national character is vehemently anti-gay, but there is a strong tradition of homosexual relations in Rekkenmark between the instructors and their students, male and female. They were divided into male and female classes to prevent horny old men from preying on young women, but sexual scandals still constantly surfaced among the faculty. While the classes are now gender neutral, this 'tradition' is still rampant. Its not openly acknowledged, though Kaius is trying to crack down on it, and it has had a widespread affect on Karrnath's military forces. Still, pregnancies are lower than the the gender integrated armies of other nations, so many Warlords believe it makes the Karrnathi army stronger and more unified. This student-teacher sexual relationship is beginning to become more common among male/female pairing with the integrated classes, but the old tradition is still very strong. Also of note, the participants of these relationships do not consider themselves homosexual. Effectively, the majority of the Karrnathi officer class could be said to be on the 'down-low', in a sense. Their eagerness to defend their heterosexuality (men and women) heavily contributes to the nations intolerance for open homosexuality.
Breland is of course more cosmopolitan than the other nations, and in the cities homosexuality is not considered very unusual. There are of course many bigots to condemn it, but the sheer number of homosexual bordellos in Sharn alone attest to how little most of the populace actually care about it. A different story in rural Breland, which is still against the idea of homosexuality, though not as extreme as Karrnath or Aundair. Not as accepting the general populace of Thrane, but you won't be attacked simply for being homosexual (Or its a rare occurrence at least). More likely than not, you will be shunned from the community.
Cyre, being the seat of Old Galifar's administration, was if anything even more conflicted than Karrnath. Galifar himself declared the practice immoral and outlawed it in his kingdom, and as the center of the kingdom Cyre was expected to maintain that image. However, with the very large concentration of intellectuals and artisans, it had a very large underground following, creating a sharp divide between the peasantry (who supported the will of their King as the absolute moral truth) and those born more privileged (the merchants and lesser nobility, mostly). Up until the destruction of the nation, this was a great dividing factor among the people of Cyre, but now that there are so few Cyrans left, it has become something of a statement against the other Nations to reject their culture and maintain the culture of Cyre. As such, the population of the Cyran refugees has a higher percentage of openly gay men and women than any of the Five Nations. There is a lingering resentment against it among the populace, but desperation to save anything of Cyre's culture has pushed this practice to the forefront, to the point that it has become fashionable among young heterosexual Cyrans to pretend that they are homosexual, as a point of national pride. This has, of course, served to strain the relations between the refugees and the Nations, and Prince Oargev, while seeing nothing wrong with the practice personally (I like to think of his as an accepting and likeable person) is trying to curtail it to help raise the general opinion of the refugees.
Among other nation, the Mror Holds mirror the Karrnath situation, but even harder (Tee hee... 'harder'). Dwarven society is carved in stone, and from the time they were exiled from the old kingdom, the Clans knew that they could not allow for homosexual practices to curtail the growth of their population. To that end, I imagine open homosexuality in the Holds is punished severely. And that, like Karrnath, it is still absolutely rampant throughout their military, though kept 'secret'. After all, its a bunch of burly men alone on a mountain. I've never seen Brokeback Mountain, but if I did then I'd make a joke about it here.
Darguun, I think, is easily the most accepting nation for homosexuality in Khorvaire, outside Valenar (possibly). The ancient Dhakaani Empire never cared about who was having sex with who, as long as you were doing your duty for the Empire and pro-creating. I like to think they may have had analogs to the Sacred Band of Thebes. I know the homosexuality aspect of the Sacred Band is disputed, but I like the idea that the ruthlessly pragmatic Dhakanni Empire would take advantage of those relationships for their morale boosting effect on battles. That tradition continuing on among the Dhakanni clans today.
Valenar, I'm not sure. Maybe the idea of homosexuality as something 'different' from the norm doesn't even occur to them. I'm fairly sure the Aerenal procreate based on the best physical match (like two great wizards having a baby with the hope of siring another great wizard) and don't tie 'love' with 'sex'. So maybe the Valenar don't have any opinion on homosexuality because its just sex and who cares. Could be interesting if a heterosexual Elf is chosen by a homosexual ancestor and expected to relive their 'deeds'. Or a homosexual Elf imitating a heterosexual Casanova of an ancestor.
The Lhazaar pirates have attitudes based on what principality you're sailing in, and same for Droaam. The Daughters don't care, it depends on what the local warlord says.
I don't think the Druids of the Eldeen would care at all. They'll marry your horse to a thornbush out there. The former Aundairians might be more against it though, but I still see it being marginal.
Q'barra needs to grow its population, so I imagine heterosexuality is encouraged, but given the political dissidents and what not crowding the place, I don't think they take a hard line against homosexuality.
Shadow Marches, I dunno. We could all be eaten by Daelkyr tomorrow (if we're lucky) so go ahead and go for it. I guess.
Hmmm. I've had gay characters in games I've run, but I never really thought much about.
I figured that since Galifar is commonly described as this idealistic and great time and all that bedroom antics were usually nobody else's concern. About the only people who would have reason for concern would be anyone looking to leave their wealth/power to blood relatives, and even then, they'd only be concerned if their son/daughter didn't produce an heir, not who they engage in recreational sex with. With enough money and authority, people in most any civilization end up sleeping with whoever they want to anyway. In fact, I seem to recall reading in the Five Nations book that King Kaius has a "harem" consisting of both males and females who additionally serve as his primary source of blood.
Considering how in human history there is a large difference how various cultures treated it (from one extreme to the other), I think it is entirely up to you. Equal sex rights, rich multinationals that openly earn money in the sex trade, polynistic religions, lack of religious persecutions, religions not being particular strict about how you live and so on all suggest it is unlikely to be an issue.
In the interests of not dampening conversation in any way, I'm not going to give my opinions on the matter at the moment. However, purely as a question of sexual identity, I will point out that unlike the Kalashtar, the gender of a Tairnadal elf's patron ancestor doesn't have to match the gender of the living elf; case in point, High King Shaeras Vadallia embodies a female ancestor. I had a male Valenar PC who decided he had a female ancestor and spent some time exploring what this meant for him.
I wouldn't imagine a big problem with homosexuality in the central five nations, aside from the possibility of Karrnath, though it might be a law there that is often broken and seldom enforced, possibly broken within the military as others have said. Aundair might have a few people who frown on these actions, though Breland (Different is just different) and Thrane (Probably not a very sexual people to begin with) probably wouldn't care in the least, Cyre might have the same problem as Aundair. That being said, there might be a problem in more rural areas.
Outside the Five Nations-... I see the Zil as having a tradition of arranged marriages due to their familial nature, such marriages might have a place within their various intrigues, though I also think there might be a large number of affairs outside of these marriages, regardless, I imagine they wouldn't differentiate between homosexuality or heterosexuality. The Mror - and Dwarves as a whole - might place more of a stigma upon such actions, though I could still see some practicing in secret. The Aereni have been said to think of love as a very informal thing, so I can't imagine them having a single problem with homosexuality, as long as those Elves continue to actually breed. Goblins-... I've no real idea on them.
If we're going into the place of the other continents: I suppose the Dragons of the Thousand and the Light of Siberys might have a stigma on homosexuality, though the Tapestry and the Vast mightn't care, Seren would vary depending on the Tribe. The Inspired want to produce more Chosen, I also see closed-mindedness as a small part of what defines the current Dreaming Heart, so just as the Inspired inspire an inferior view of the other races (Excluding Changelings), they might also inspire such a view of homosexuality. I think homosexuality would actually be rather common in Kalashtar, for both similar and other reasons. Xen'drik is real uncertain.