Are mosquito bites harmful or just annoying?

Mosquito bites aren't "harmful" as the game has no HP system and you cant die or anything like that.

However, as stated by Alex, it prevents you from catching them. Residents may also have a new dialogue with you. Getting bitten by one also attracts more of them (creating an annoying cycle of stings)

You can avoid mosquitos rather easily, they have a short 'aggro' range and you can hear them coming (high pitched buzzing). They are also very slow moving, so they are easy to catch (once you spot them that is).

In short, they are just annoying. They only appear Summer evenings, though, so be a bit more careful during this timeslot.