As far as I know, The Golden Girls has Dorothy and Blanche writing fake love letters to Rose, posing as Rose's nonexistent suitor. Eventually, that led Rose into mistake the one with the same name as that "suitor".

In Cheers, the guys wrote phony love letters to Carla, posing as her dream suitor, causing her to almost turn down an actual person responding to her newspaper ad.

Are there any other sitcoms having characters do romantic charades to their friends?

The Simpsons. Bart posed as a latin lover/pen pal to cheer up Mrs. Krabapple.