Some of the funniest songs I've ever heard are:
youtu.be/hnzHtm1jhL4
youtu.be/cmjrTcYMq8M
youtu.be/ASovlGbbBm4
youtu.be/BxsQ9vESX0Y
youtu.be/SMhwddNQSWQ

Definitely that Tenacious D song: www.youtube.com/watch?v=2Jvgbe… but it was catchy back in the day!

"Trippin'" by Gravediggaz always seems to make me laugh for some reason and especially the part that goes "Down the corridor was old fat Ned" and there's more to it, which might be too dirty to mention... LOL!

The Beatles song "I Am The Walrus" for obvious reasons...

erm....weirdness - www.youtube.com/watch?v=vs1CA5…
Old Skool TMNT -  www.youtube.com/watch?v=y9eKwF…
Im liking Miley Cyrus's new voice - www.youtube.com/watch?v=hrydcs…
Hi ho, Hi ho, It's off to work we go - www.youtube.com/watch?v=qkTRam…

Poo Pants- Cyriak 
What Gareth Brown Says- Mclusky 
Jump You Fucker Jump- Derek And Clive
He bite (on Rather good.com)

Miracles by Insane Clown Posse. It's a "So Bad it's Good" song in my opinion. And Gangnam Style is kind of a mix of funny and good to me, as well as Hangover with Psy featuring Snoop Dog/Lion. Psy generally makes his videos fairly silly, and Snoop just adds to it.
Also anything by John Lajoie or Stephen Lynch.

This version of Scent by Die Krupps. Great band and good song, but that fake American accent is impossible to take seriously and the vocalist is trying too hard to sound "metal."

This song is so bad its funny its yo ho blood on the dance floor www.youtube.com/watch?v=mYlYR9…

GoRemy is a rapper on YouTube, and his works are usually pretty funny.  I memorized the rap in the following link, but his "Slow Jam" is really funny.  youtu.be/Yo-_Y4aL1uo

"Strawberry Cake" is a funny song by Johnny Cash.

Tom Lehrer is a satirist who wrote the "Elements Song," and one of my favorites of his songs is "We Will All Go Together When We Go."

The song that the Element Song's score is based on is a satirical song, called "The Major-General's Song," from Pirates of Penzance.

A song my sister showed me a few years ago named "I'm an Ugly Girl" which is supposed to be a parody of a song named "I'm a Barbie Girl" www.youtube.com/watch?v=l8MTy8…