To me, it's always "sit close", never "sit closely". Perhaps you could provide some context?

both can be used but 'sit closely' does sound better. Sit close to each other could be other way of putting the first one in right way.

You're confusing the adverb closely with the preposition close. Closely is a manner in which you sit, same as quickly or silently. Close is where you sit, same as below or outside.