A few people have suggested Stella already, but I'll further enforce that suggestion because it's a great show. I don't really know any other shows to throw out there, but you should check out the movie Wet Hot American Summer. A few of the people involved in this show are involved in that movie, and it's overall really funny (got terrible reviews though).
Also, on an unrelated note, check out the movie Kung Fu Hustle. There's no relation between this and Childrens Hospital and it's a different type of humor, but I figured I'd throw it out there anyway.
I want to strongly second Wet Hot American Summer. I bought the dvd around the time it came out and told all of my friends to rent it. After renting it they bought it and told their friend to rent it. I butterfly effected a lot of sales for this movie I'm sure. It's worth a purchase, you can watch it over and over because it's so quotable. The dvd has an optional 'fart track. Basically random fart noises are inserted throughout the film. It's genius.
I like Eagleheart, some episodes are fantastic(death-punch), some are ok. NTSF is meh. For Flight of the Conchords fans you might want to check out Eagle vs Shark, a low budget NZ movie with Jermaine.
I was a little let down by Wet Hot American Summer. It was a good film but not quite as good as I had been led to believe. I still recommended it to some friends though. It's worth watching after a few drinks.
Some shows similar to Childrens Hospital would be 'Campus', 'Darkplace' and 'Greenwing'. Also check out a film called 'The Ten'. It's the same group of actors and writers. Some good sketches.
"I saw the best minds of my generation destroyed by madness"
Definitely The Ten! A lot of the same characters even (for example, Ken Marino is Glen in both The Ten and Children's Hospital and Louis LaFonda is in both too ).
If you want to go old school, I'd suggest "Police Squad!" (hilarious but short-lived), "Fernwood 2-nite"/"Mary Hartman" and "Soap" -- maybe "Action" (although not as absurd). And skit comedies like "SCTV" and "Kids in the Hall".
More current shows would probably be "Archer" (very funny) and "Community".
"Animal Practice" on NBC reminds me of this...similar set up...comedy in an environment usually geared towards drama. But Children's Hospital is so much funnier.
Look Around You (here's the episode on "Water" http://www.youtube.com/watch?v=5lDSjuo67LM)is a great spoof on those cheesy, poorly-made instructional videos you watched in your science class.
Have to second some of my favorites: Archer, Arrested Development, Flight of the Conchords. Didn't love Frisky Dingo, but I might have to give it another chance.
Sadly, every Show I love gets canceled. Hope this and Archer stay strong.