Yes. If you turn off your filter for a few days, you will lose your good nitrifying bacteria, have to cycle the tank again when you restart it and your Ammonia > Nitrite > Nitrate cycle will stop. You will end up with an ammonia spike which could seriously impact your sick fish and prevent them from fighting the infection efficiently, or even kill them if it's that severe.
Never turn your filter off for longer than it takes to clean or change your filter media.
Just take out the carbon and turn it back on. As
kno4te
said, keep the power head on.
Whack up your temperature to the max your species can handle, it helps kill the ick, but watch for signs of gasping and increase surface movement if they start gasping. Temp increase = less oxygen = potentially more surface agitation needed.