The famous joke
>"Three tomatoes are walking down the street -- a poppa tomato, a momma tomato, and a little baby tomato. Baby tomato starts lagging behind. Poppa tomato gets angry, goes over to the baby tomato, and squishes him... and says, 'Catch up.' "
Instead of "Catch up", they translated "Ketchup".
But as "Ketchup" doesn't sound like the German "Catch up", it sounded like an anti-joke.
If I recall correctly, the works of Edgar Allen Poe are famous for having liberal but high quality translations in German. Song writers also sometimes do this when "translating" a song to another language.
e.g. another joke like this Did you hear about the country with the fastest growing capital? It's Ireland - every day it's Dublin.
I can't think of any way how to translate this joke so it would be funny in German. But I have a feeling that these word play jokes aren't used much in German speaking countries anyways. Most of the humor is about people acting different or surprising.
The closest equivalent I can think of are sentences where the joke lies in using a different word at the end than the (likely, extremely crass) one the audience expects. These tend to work better in German than in English because of the sentence structure, I think.
EDIT: Example of a dad joke pun: "Was sagt ein Sachse auf dem Weihnachtsmarkt in New York? Ä Tännschen, please." -- "What does a man from Saxony say at the Christmas market in New York? A fir, please."
The joke is that "ein Tännchen", "a (small) fir", when pronounced with a Saxony accent, sounds a bit like the English word "attention". The joke works because most Germans know the phrase "attention, please" from movies, vacations and/or multilingual announcements. I believe this particular joke was invented some time in the 60s or 70s and ceased being funny about two seconds after, but that wouldn't stop "your dad" from using it.
I'd say that most of these double meanings are hard to translate. At least if you compare German and English (the two languages I am "proficient" in). If you want this in a more modern context I would be completely unable to translate Dendemann lyrics to English so it would still be lyrical or funny.