TV Theme songs that were played during the episode not just the beginning and ending credits.

Dennis the Menace
The Monkee’s
eight Is enough
He-Man and The Masters Of the Universe (80’s)
She-Ra and The Princess of Power (80’s)
Who’s The Boss?
Growing Pains
Just The 10 Of Us
Amen

Coach often used its theme song during the episode.

The theme music of Cannon is played in the episodes, at a slower speed.

part of the TAXI theme was played in a few episodes.

Part of the Brady Bunch theme is played, at a slower speed, between some scenes.

This isn’t exactly answering the question, but when Valerie Bertinelli guests on Boston Legal, as William Shatner’s character watches her walk away the One Day At A Time theme song is played.

Sometimes the Happy Days theme is played during scenes.

lol! True!

The King Of Queens played Mr.Belvedere theme song.
Roseanne played Leave It To Beaver theme song.
Roseanns played The Patty Duke Show theme song.
Roseanns played Bewitched theme song.
Full House played The Brady Bunch theme song (end credits)
Full House sung Three’s Company theme song
Just The 10 Of Us played Just The 10 Of Us theme song
Growing Pains played Growing Pains theme song
Growing Pains played Just The 10 Of Us Theme song
Newhart played The Bob Newhart Show theme song
Home Improvement played The Monkee’s theme song

Little House on the Prairie: theme music pops up at times (music had also been used for late episodes of Bonanza).

Leave it to Beaver (slower version)

Dinah Manoff sang the EMPTY NEST theme in the tag scene of the Season 7 episode "The Courtship of Carol's Father" which had a big scene that takes place in a piano bar.

Also, Hoyt Axton guest-starred in a FLO episode and sang Flo's Yellow Rose (the series theme song which he also performed) to the characters at Flo's bar.