Like many people, I'd imagine, the first version of this song I encountered was the one featured in the first episode of Animaniacs.
(if the video is taken down by the time you read this, just do a google search for "Animaniacs Monkey Song" or better yet pop in your Animaniacs Volume 1 DVD!)
But I was pleased to find the original version was also available on YouTube:
Some people might call the Animaniacs version a "parody" of the original, but it isn't really... it's just the Warners and Dr Scratchansniff doing their own version of it. The tune is the same, and most of the lyrics are very close. The instrumental breaks are in the same places (accompanied by scenes of other Animaniacs characters doing their schtick), and some of Belafonte's interjections are repeated by the Hippos. "Play that thing!"
It was a surprise that the Belafonte original has only one monkey causing him grief... I guess we naturally think of monkeys coming in trios? (notice the poses the Warners are doing at 2:27!) They just upped it to three monkeys for the Animaniacs version because there are three Warners.
Verse 1: First two lines very close, original has a repeat of the lines, Animaniacs version has two new lines which are specific to the characters and setting. (in fact, the Animaniacs version avoids line repeats a lot more than the original, with more lines rhyming with "Don't know what to say the monkeys won't do" in the chorus parts.)
Verse 2: This is the most different from the original version. Belafonte has the monkey letting his girlfriend in and pouring her a glass of gin. Scratchandsniff has the Warners doing a dance for the Nurse and pouring biting bugs down his clothes. The only similarity is that the end of the second line ("and what do you think?" vs "and what could be worse?") are quite close.
Also, a common theme in the original is the monkey copying Belafonte's actions: "I do X, monkey does X too." The only time something like this appears in the Animaniacs version is here, and it's the other way around: the Warners make Scratchansniff itchy so *he* copies *them.* "Monkeys dance, then I dance too!"
Verse 3 is almost exactly the same, but with the Nurse in place of Belafonte's girlfriend.
Verse 4: Also very close, but this is where the exaggeration of the original starts. Belafonte's "cabinet" was laid to waste, but Scratchy's entire bathroom is. Belafonte has to shave with toothpaste... but the Warners actively shave Scratchy's *head* with the stuff.
Verse 5 continues this exaggeration. Belafonte brushes his hair with a shoe-brush, the Warners use one to shine Scratchy's (shaved in the previous verse) head. Belafonte "almost" goes down (the toilet, presumably), while Scratchy is not so lucky.
Verse 6 is even closer to the original than verse 3... which is quite easy to believe, as trying to make a stew out of the Warners isn't exactly Scratchy's nature. (He's not an Elmer Fudd type hunter/predator, but more like a long-suffering parent or teacher when we usually see him) New to the Animaniacs version is the ensuing chorus which puts Scratchy in the stew himself.
So, as you can see, most of the lyrical changes were small variations to make the Warners more actively annoying than the original.
Say... if you're interested in more Animaniacs commentary, check out Mike R's incredible in-depth blog Hello, Nice Warners!