No, they do not need to be removed. Strawberry plants are only productive for a few years (2-3) before the mother plant needs to be replaced (that's what the runners are for), so it doesn't make much sense to wait. Doesn't really affect the plant one way or the other.

I've read that in many sources, too. I never do it, though -- I'm too anxious to get the strawberries! I never noticed poor plant vigor or anything like that. Good to know it's not really necessary. I grow ever-bearing types, IDK about June bearers.