A better question would be "Do comedy songs get boring?". I must be fair, I love some of the 'songs' of Les Barker (Reg: The lonely glow-worm. Dachshunds with erections can't climb stairs). And, although his poems often to seem to be set to well-known tunes (and are often parodies thereof) he doesn't actually sing them. It's more of a poetry recital.
If you get a chance to be at one of his evenings, you will likely enjoy it. But as the evening goes on he gets requests for certain items which are part of his regular repertoire, and people join in with the punchlines. These are just as amusing as the first time you hear them, possibly more so. Les is not only a writer, but quite clearly a 'performance artist'. Just reference to his performances seems to negate the original question.