More specifically, puns in style of Manzai, a minigame in the original Japanese version of Rhythm Heaven Fever, you can watch it from here (Japanese) https://www.youtube.com/watch?v=kdZtxJnbpbU
Here are few of the examples from the game. Just take note of the transcription. Full List.
布団が吹っ飛んだ (futon ga futtonda, The futon blew off)
ミカンがみっかんない (mikan ga mikkan'nai, I can't find any mandarin oranges)
Basically, in a language of your choice, come up with a sentence with similar sounds repeated. Feel free to make more than one!
I'm curious if lojban can have actual puns given lack of ambiguity is one of its design goal if I recalled correctly. I don't know really much about log langs I think its possible
There are some puns, I just can't think of any for this specific style.
One of my favorite Lojban puns is the word for joke words, {.u'ivla}. It's {.u'i} "haha" plus {vla} "word" (as combining form), and looks pretty similar to the word for loanword, {fu'ivla} "copy word" (fukpi valsi).