Mysterious Mysteries (Dib’s favourite show on Invader Zim)
Well, if we’re going there…the full title of the show is “Mysterious Mysteries of Strange Mystery”, as given in a couple of shows.
Invader Zim also features “The Scary Monkey Show”, which Gir watches whenever possible. And Dib’s dad hosts “Probing the Membrane of Science”.
Smartline, with Kent Brockman.
Also, Eye on Springfield, also with Kent Brockman.
And from Kim Possible, The Fearless Ferret, starring Timothy North (voiced by Adam West).
Actually, The Money Programme is real.
George Burns had a very strange TV show within his TV show.
He would be inside his house set and the TV would be on and he would see his neighbor approaching and go to answer the door.
It was very clever self-referencing. His “part” of course was that he was playing himself at home but with asides to the audience.
Oh, yes, that’s the one I first though of… 
What, you remember The Happy Homemaker with Sue Ann Nivens but you forget Chuckles the Clown???
Studio 60 on the Sunset Strip
TGS (The Girly Show) with Tracy Jordan
You Bet Your Ass, from Contact
When I first read that novel, I laughed out loud. As if a show would ever have a name like that! These days no one would bat an eye.
This is a movie, not a television program.
SCTV:
The Bobby Bitman Show with Sammy Maudlin
The Sammy Maudlin Show featuring Bobby Bitman
Then there’s Real People from WKRP.
I always come up mostly blank in these things. Although I do remember (with a little looking up for he right title) Mysterious Mysteries of Strange Mystery from Invader Zim.
Too bad we’re restricted because I also have a movie that people would assume is a fake made-up one but is actually a real movie and some radio shows that are completely fake.
Invitation to Love
Would You Buy it for a Quarter?
Eerie, Indiana (in the episode “Reality Takes a Holiday,” where Marshall discovers he’s in a TV show named Eerie, Indiana)
Didn’t Captain Qwark have a TV show in the Ratchet & Clank video game?
There was a Monty Python episode involving a television character called “The Bishop,” which was a parody of The Saint. The running gag was that whenever anyone would say “The Bishop,” the show’s opening credits would run.
But, if you pay attention in that bit, the first time the credits repeat, we hear the comment “this is the part where we came in”, and see a couple of the Pythons leaving a movie theater.
SCTV:
The Johnny LaRue Show
Monster Chiller Horror Theater with Count Floyd
Springfield also has “New York Nookie” and whatever show The Bumblebee Man appears on.
I’d Buy That for a Dollar!