In my experience, ‘I’m not being funny, but...’ is used to preface a statement that might give offence or that is more assertive or direct than the speaker is entirely comfortable with. As such, its function is to soften the subsequent statement, or at least signal that the speaker is aware that what he/she is about to say might not go down well. Whereas ‘Da verstehe ich keinen Spaß’ intensifies the statement to which it refers, right?
I don’t quite understand the problem with ‘that’s no laughing matter to me’ – I think it’s a valid option that conveys the general sense of the German in certain situations. Here are a few other natural-sounding sentences that might be used in the situations mentioned:
I’m usually pretty laid back, but I find it utterly infuriating when people smoke while I’m eating.
As an apprentice, you are expected to arrive on time, Meier. We have a zero-tolerance policy on that.
The police tend to be less strict about low-level speeding, but they have a zero-tolerance policy when it comes to running red lights.
Please don’t play with your food, Maja. I’m serious.
Maja! Stop playing with your food. Don’t make me say it again.
But maybe I’m misunderstanding something.