Yeah if DH has it then he could pass it on to you. Maybe get DH to do something to up his immune system to help prevent and get rid of them, so you don't keep passing them to each other.
Definately mention it to your doctor, the doctor will give you ideas on what you can do to combat them, and maybe ask what DH can do.
They probably won't burn it off during pregnancy, they wouldn't burn mine off, and stay away from the stuff they sell at the chemist to get rid of warts, they aren't recommended for use during pregnancy.