Short answer: No.
By countably infinite subset you mean, I guess, that there is a 1-1 map from the natural numbers into the set.
If ZF is consistent, then it is consistent to have an amorphous set, i.e., a set whose subsets are all finite or have a finite complement. If you have an embedding of the natural numbers into a set, the image of the even numbers is infinite and has an infinite complement. So the set cannot be amorphous.