As Gary says, it's a difference of timing. This creates a big difference in meaning.
"Pay me to not do anything"
This means that the payment, or at least the deal, comes first. It sounds like a kind of bribe. Perhaps someone gives you money in order to persuade you to do nothing (for example, keep your silence, not take legal action). The split infinitive is deliberate - it is important to keep the 'not do' together. The idea that you should 'not do anything' is key to this sentence.
Two points to note about the 'to + infinitive':
1) It often has a future meaning. When we see this construction, our instinct is to assume a forward-looking scenario (as in plan to, decide to, intend to, hope to). The 'to' carries us forwards.
2) It is fine to put 'not' between the 'to' and the verb. In this case, it's necessary for the meaning. The 'don't split the infinitive' rule is a piece of 18th century nonsense. It's a Latin-based convention that should never have been applied to English.
"Pay me for not doing anything"
Here, we presume that the payment comes afterwards. This sounds like someone has a job where they do nothing but still get a salary at the end of the month. Compare this with similar constructions "Thank me for doing something" or "Criticise me for doing something" - first comes my action, then your response.
As you pointed out in your response to Gary, in Italian you would need to use a perfect construction ("Pay me for having done nothing"="Pagami per non aver fatto nulla") to get over the idea the idea of a past action. In English, just the 'for'+ gerund does this job.