I think some of the answers/comments need to be pointed out as flawed.
"The liberal media", doesn't make sense because TV is a for profit business and if a conservative comedy hour drew enough listeners, it would be aired, in fact, given that conservative listeners tend to be quite loyal, I'd go so far as to say TV stations would fight for the rights to air a conservative comedy hour that drew a sizable audience. Dennis Miller doesn't have a larger audience because he's not very funny. I would add, the Daily show would make fun of a liberal politician in a heartbeat if it made good TV. Would Dennis make fun of conservatives? If he would, I've never heard him do it.
"Punching up vs punching down", in one of the comments. While I absolutely agree with that statement, Obama was president for 8 years. George Soros is a billionaire. Nancy Pelosi was speaker of the house and a powerful congresswoman for decades, successful liberal actors who speak out are, rich and successful. Making fun of any of them IS punching up. It's incorrect to think Republicans/Wealthy are the only ones who are "up" and the only subjects to comedy.
And I tried watching that awful video by Prager U, and while some of his comments were fair, he doesn't understand comedy. Yes, there's political correctness in today's world, but not all comedians yield to political correctness. Many defy it. Political correctness might raise some objections, but some comedians tell the joke anyway and what's funny is still funny, perhaps even more so when it's edgy.
I think all of those answers miss the mark.
What it comes down to is this. Funny is hard and funny is rare. It takes a precise set of criteria to make something funny and it's not easy to do. I've heard it said that Hitler jokes are never funny, and I've never heard a funny one, so, I'm inclined to agree, but John Cleese pulled it off in Faulty Towers. That said, Cleese was brilliant, and they found a clever angle, a guest at the hotel was sensitive, so "don't talk about the war" and one thing lead to another.
But being funny is hard, especially for 4 shows a week.
The essence to this question can be summed up like this. Why is Sarah Palin comedy gold but Nancy Pelosi isn't. Sarah's clumsily worded moments invite mockery and I'm tempted to say, funny stuff, while Nancy's are simply cringe worthy, but not the material of real humor.
Sarah is comedy gold, Nancy isn't. Why is that? Basically this is just re-asking the question, but it helps narrow down what I think is the answer. The criteria for funny is easy to miss and hard to hit. There's a very narrow range where the comedy actually works, draws a sufficient audience and most important, makes people laugh out loud, cause that's what the audience is there for.
The subject is either inside the Goldilocks zone or they aren't and Nancy, or Obama, or other democrats, for better or worse, were never comedy gold. Some of Nancy's comments invited enormous reaction and repetition, but she was never quite right for comedy.
Perhaps, it's not the mistake that's funny, it's covering up the mistake that invites the comedy. That's the Lucile ball principal. It's not getting upset that's funny, it's pretending you're not upset when the audience knows you are.
That said, dissecting what makes something funny is far from a perfect science, and when I tied dumb to funny, people got offended but in a sense, dumb can be funny. There are in fact, several types of funny, some articles say 9, 10, 11, I saw one that listed 20. I still think the answer to this question lies in looking at specifically what's funny and working backwards, because there's a narrow window that actually is funny.
That's my answer. It's not about making big assumptions but about reverse engineering why things are funny. Start with the skit and work backwards and find the elements of humor. The Daily show appears liberal, though I maintain, they would make fun of a liberal politician in a minute if they thought it would get laughs.