They annoy them as badly as they do us. I try to keep them at reasonable levels with a fan on the birds.
Mosquitoes carry avian pox. There is a vaccination for it that the feed store should carry. If they get it, it will just run its course and then they will be immune for life, or so I have read. It causes black lesions on their combs and wattles. It is viral and there is no treatment for it.
I didn't know about the vaccination in time, so wound up treating the lesions with some Betadine or Neosporin to prevent a secondary infection. After a few weeks, the lesions simply disappeared.
Occasionally, it can cause lesions inside the mouth and respiratory tract, and this can be serious or fatal ("wet pox.")