Mosquitoes can and DO carry diseases that can harm just about any animal they bite. Fowl pox is the one that ones hears about most, and if the bird becomes infected internally, death is very common. Infection via the skin is ugly and uncomfortable fo rthe bird, but rarely causes death.
However, mosquitoes also carry other diseases, such as various types of encephalitis, and those can definitely kill.
Remove standign water, spray the area with a good mosquito killing product (and preferably one that will also kill eggs), cover the openings of the coop and run with insect screening, use mosquito repellant on the birds.