I find this idea pretty appealing, so I occasionally mention it to friends of mine. Somewhat to my surprise, many of them don't quite see the point; this use of humor isn't appealing to them at all. Well, fair enough.
Russell T. Davies sure likes the idea, though, and his use of it reached its apotheosis in the fourth and fifth episodes of the new series of Doctor Who, the two-parter Aliens of London / World War Three. It might not be possible to enjoy this episode if you don't agree with Douglas Adams about the effectiveness of this technique.
It may be useful to give an example. After the spaceship crashes, an 'alien' is found onboard and brought to the mortuary. Important people look at the corpse and look shocked and disbelieving. Then a single doctor is in the darkened morgue, we hear ominous pounding from the locker, the alien is alive after all and gets out, etc., and finally we get to see the alien, and it's absurd -- a sort of pasted-together two-legged pig. (This is the point at which viewers laugh, or possibly are annoyed.) The creature is killed and the Doctor declares that it is, in fact a pig, but one that's been modified so that people will think it's an alien.
So. A couple of things going on here. The main point is that the pig-creature looks absurd (as was the case with many old-school Doctor Who monsters), but it's supposed to look absurd, and the fact that it looks that was is important to the plot -- the real aliens created it (presumably in a hurry) to take the heat off of them. Also, I like that the initial event, where the Big Important Guy looks at the alien (but we can't see it) and can't believe what he's seeing, initially seems to mean one thing (it must look really alien and scary!) but turns out to mean another (it looks totally absurd and unbelievable!).
Another example, and one which is probably hardest for people to swallow, is the whole business of farting. The fact that the Slitheen (when in human form) fart constantly is an obvious slapstick comic touch, but it serves several purposes as the story goes on. First, when we don't know that the minister is an alien, we think it's a silly piece of slapstick intended to characterize him as a bumbling oaf who's in over his head. Then we realize that he's an alien and that the farting serves to identify the aliens in human form; this still seems a little thin, but there were one or two points in the series when a character farted and I actually felt a little chill -- the connection between 'farting' and 'alien' was effective, at least for me. Finally, the fact that the farts smelled a certain way allows the Doctor to determine their planet of origin, their composition, and the way they could be destroyed.
There are other examples of the use of this device in the episode. But, in fact, the entire two-parter has the structure of a joke that is taken seriously. The initial absurdly dramatic event, where a spaceship crashes out of the sky, runs into Big Ben, and lands in the Thames, turns out to have been deliberately engineered as an absurdly dramatic event, specifically to get the attention of the world, divert attention from an alien base, and set the stage for World War Three. So this idea of 'humor taken seriously' takes place on multiple levels with different degrees of importance throughout both episodes, to the extent that I think it must have been quite intentional. (No other episodes from the season make use of the device in this way.)
(I also find it interesting that one of the places in the episode that I think does fall flat is, basically, a standalone joke. It's the bit where the Doctor is backed up against a wall by soldiers who have been ordered to shoot him dead, and he explains that when you've cornered someone against a wall you should make sure that the wall doesn't have an elevator built into it. The elevator then opens and he steps in and makes his escape. It's a throwaway joke, and not a horrible one, but it doesn't make sense (the soldiers have plenty of time to kill him both before and after the elevator arrives) and serves no greater purpose. If it were in keeping with the rest of the episode it might not seem to make sense initially but in the course of time it would turn out to be completely logical.)
The way that the episode makes use of Douglas Adams's idea, and makes it work (at least for me), is one reason why I really like these episodes of the series. Although there are one or two episodes from the season that I also think are quite excellent and might even like a little better, these episodes were the ones that convinced me that the series was really hitting its stride.
sunburn can be held to be partly responsible for me writing this, since he threatened to give a cute robot an MRI if I didn't!