Mana abilities don't use the stack. So while your Mana Web
will trigger when a permanant becomes tapped for mana, that mana is already in the player's mana pool by the time that trigger is even put on the stack. Furthermore, the trigger can be reponded to (it does use the stack), so the player can just produce more mana in response (triggering Mana Web
Also, mana abilities can be activated while casting a spell or activating an ability and when an effect asks for a mana payment. In these cases, the triggers from Mana Web
are put on the stack after the spell is cast, the ability got activated or the effect has fully resolved.
In essence, the Web can force an opponent to tap his permanents for mana
all at once, creating more mana than needed for whatever he's doing, or lose the ability to produce mana with them this turn. So he has to do all the other things he wants to do, that require mana, during the same step/phase. Counterspells are still possible, but likely only one each turn. He can produce more mana for another counterspell, but you can just wait for another step/phase to cast your next spell, when his mana has vanished from his pool.