I realize this is coming way after the fact (as I’m still way behind), but I wound up diving deep on this while catching up. I somehow missed some of the followup, so I’m definitely restating what other people have said already, but hopefully the breakdown is helpful to someone at some point.
The usage of のに here is a straightforward application of the grammar point のに (JLPT N4) | Bunpro, OP just missed a stealth clause-breaking particle hidden within “ちゃ”, so they were mis-parsing the sentence and winding up with a translation that doesn’t make sense, like this:
あれは / that (the button, and the topic)
用もないのに / despite having no purpose
押しちゃだめ / it’s bad to push
“Despite lacking a reason, pushing it is bad”
Remember, though, “ちゃ” is a casual way of saying ては (per てはいけない (JLPT N5) | Bunpro). Imagine it re-written like this:
あれは用もないのに押してはだめ
Now it’s more like:
あれは / (Doorbells + topic marking particle)
用もないのに押しては / Pushing without a purpose
だめ / is bad (you must not)
“You mustn’t ring doorbells for no reason.”
(Interpreting the verb “push” as “ring” for this context.)
The one part I was a little fuzzy on is the “なの” at the end of the sentence. However, I was able to find a reference to it in Bunpro vocab (なの (JLPT N1) | Bunpro) and in Jisho, it’s translated as a colloquial, female-speaker, sentence-ending phrase for asserting things.