The Young Ones.
Although I don’t think Special Patrol Group accompanies the rest of the cast.
The Partridge Family.
Didn’t Drew Carey sing “Moon Over Parma” for his sitcom?
And he was called the “Balladeer,” I believe, not the narrator.
Does “Animaniacs” count?
On the original Battlestar Galactica, Patrick Macnee was the voice of the Imperious Leader and he also played the recurring role of Count Iblis. And I think his was also the voice that spoke the introduction during the (long version) of the opening sequence.
Linda Lavin sang the theme There’s a New Girl in Town on the show Alice.
Yes and no. The theme was recorded by a group of studio musicians and hired background singers informally known as the Wrecking Crew, although David Cassidy (Keith Partridge) did convince the producers to let him sing.
I’ll give it credit, though, since Micky Dolenz was the only Monkee to sing on their theme song.
How I Met Your Mother did it one episode. I wouldn’t be surprised if Psych did, they liked to play around with their theme song.
Dennis Waterman also sang the theme to Minder in which he played the titular role (“I could be so good for you” was the title of the theme song, I think).
Mad About You
Paul Reiser is just too damn talented.
I don’t think any of the regular Psych cast members ever sang the theme song, but Curt Smith of Tears For Fears appeared in an episode as himself and performed the theme song.
Reiser didn’t actually sing it, though. Andrew Gold did. Then Anita Baker. Reiser played keyboard and co-wrote it, though.
ETA: RE: Psych, the theme WAS performed by the Friendly Indians. which was the band the show’s creator was in.
…he has a habit of singing the theme tune.
Mystery Science Theater 3000. Joel, then Mike sang most of it with a few bits from other cast members.
21 Jump Street.
NBC was absolutely right. Way way over the top campy.
Also, New Girl theme by Zooey Deschanel.
Why on Earth would they do that? 
“Wrote the theme tune, sang the theme tune…” 
Does the theme song to ***The Muppet Show ***count? It was sung by all the actual puppets/puppeteers who starred on the show.