Nixon’s Head, FTR, is played by Billy West. (Fry, Zapp Brannigan, Professor Farnsworth, Zoidberg, and a bunch of secondary characters - including several other Presidential Heads…his Clinton is way better than his Nixon.)
Voice actors who can mimic get a lot of call to use that talent. Celebrity parodies, filling in for other actors who’ve had to take time off/quit/have been fired/passed on, sequels and spin-offs where the original actor decided not to sign on… I can’t say for sure, but I rather doubt that there are a significant number who make a career just out of mimicry, though.
One thing that annoyed me about Madagascar was that the penguin leader was clearly supposed to sound like Phil Hartman, but the producers didn’t want to seem to acknowledge it.
I once wrote some dialog for a minor character in a videogame with Patrick Warburton in mind, but when it came time for casting, we couldn’t justify his rates. So we found an actor who would work for scale who could do a passable Warburton knock-off.
I think this happens fairly often. A role is created with a well-known voice as the model and then the casting director is told “Find us someone cheap who sounds kind of like X!”
This is what I was going to mention. There’s a pretty fine line here, but if you’re doing it in such a way as to make people think the speaker is Mr. Famous So-and-So, then you’re over the line.
It’s an interesting and somewhat unsettled question. We briefly studied the case of the Bette Midler Ford commercial back in law school (warning: PDF). In a nutshell, Ford wanted to use Midler’s version of the song “Do You Wanna Dance?” in one of its commercials. Midler refused, saying that she didn’t do commercials. So Ford hired one of Midler’s backup singers, instructing her to do her best Bette Midler impression.
The case bounced around the courts for a while, but eventually a Federal appeals court found that Midler had what amounted to a property interest in her voice. However, the decision was highly fact-specific, which effectively means that it’s unclear how much the idea applies to other situations.
A friend of mine has a brother who is a voice actor. He gets a lot of work because he can do a passable Don LaFountaine and others, without the premium rate. Which of course means he is wealthy rather than extremely wealthy. He also does a lot of cartoon and video game work as well. Being able to do impressions isn’t as valuable on its own, but the kind of person who can do them would typically be capable of voicing a wide range of characters often within the same show.
As for using impressions of celebrities in commercials, I once saw one with a Don Knotts impression, with “celebrity voice implied” at the bottom of the screen the entire time. I don’t know if that is enough to prevent legal action.
Please check out the Robot Chicken clip. It’s not clear if it’s Josh Robert Thompson or Tom Kane, but it really is an amazing likeness. Here’s another one. I couldn’t find a decent link to their March of the Penguins spoof.