Can mosquito bites cause fever?

No: Mosquito bites, by themselves, should not cause fever. Bites become "inflamed" commonly (not infected) and can be red and a bit swollen, but again no fever. True "infection" is quite uncommon. That said, mosquitoes, in certain parts of the world, can transmit all kinds of infections which can cause fever: malaria, dengue fever, yellow fever, are all spread by the bite of an infected mosquito.

Skeeter syndrome: One type of localized allergic reaction to a mosquito bite is called Skeeter Syndrome. Reaction to polypeptides in the mosquito's saliva can cause inflammation (red, swollen, tender), swelling & sometimes fever.