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It would only be infinite if an opponent didn't lose the game from having zero life. Otherwise after each cycle (guildmage's second ability), the opponent will be checked to see if they still have life and the cycle would continue until they lose as a state based action for having zero or less life. If they have some means to prevent losing from having zero life then I think it would cause a mandatory infinite loop (there are no choices to stop), resulting in a draw.

If the opponent has the means to place the same cycle under his control above yours in the stack, then his cycle will remain above yours in the stack and his would continue to resolve before yours. All of the relevant abilities would have needed to resolve (so the ability of the guildmage must have resolved at least once setting up the delayed triggered ability, and the top object on the stack is life gain for the player who will win or life loss for the player who will lose). If an ability causes both players to gain life simultaneously, or both players to lose life simultaneously, then APNAP means the non-active player will be the player with their cycle on the top of the stack, above the active player.

The only means to interupt the cycle would be for either player to destroy the enchantment as far as I can see. This means that the either player (in the case that they personally have a means to destroy the enchantment) can choose to either draw the game, or interupt the loop and play on.