Does ChaCha20-Poly1305 need random nonce? RFC 7539 says:


  A 96-bit nonce -- different for each invocation with the same key


It doesn't explicitly say whether it has to be random or not. If we use an ever incrementing counter. Would that be secured? Generating cryptographically secured random number is a slow process.
No, it doesn't need a random nonce.  Yes, if you use an incrementing counter, that works.

As the RFC says, the only requirement is uniqueness; as long as you make sure that each nonce you use is different, you have met the requirements - an incrementing counter does that quite nicely (and, in fact, is commonly used in practice)