What are some grammatical features found only in modern languages?

This really depends on how you think about grammar. Your question sort of implies that it's primary morphosyntactic features you're after.

And here, it's more likely to be the other way around. Things like dual and evidentiality seem to be disappearing as Trudgill has argued in a recent book (useful video overview here https://www.youtube.com/watch?v=rjy1CkH1FOE). And it is not really a question of 'modern' vs. 'ancient' but rather related to size of community and amount of contact. At least, that's Trudgill's argument. I don't subscribe to it fully but there's enough data there for the answer to the general question.

The whole question of 'modern' language is really problematic. The anthropologist in me would say that it's hiding a world of technological and developmental normativities. Meaning, that it thinks that somehow there's a directionality to language development and that direction is towards our language, the natural way of being. There's a world of argument and data in anthropology to show that we need to consider even supposedly isolated communities to be modern - and I think this applies to languages as well.

If you step outside of morphosyntax, however, you could talk about a lot of language uses that require certain linguistic tools. One of these is writing which requires stylistic, cohesive and other para syntactic devices not likely to be found in languages without a written culture of a certain kind (ie. it's not just about writing spoken words down). So many arguments have been made about hypotaxis being a 'modern' invention, etc.

I think there's good reason to be skeptical about any such claims as absolute because of the fundamental lack of evidence from the past. However, there's probably enough there to start thinking about some broad tendencies. I'm not aware of any critical synthetic account of this, though.