As far as I know, GMP doesn't many any attempts to be generally resistant to timing channels. This year's Bleichenbacher variant[1] specifically calls out GMP's modular exponentiation API (which is what you'd use for RSA) as being susceptible to timing.
In reality, RSA signing with blinding will make any implementation (including those based on GMP) resistant to side channel attacks, targeted at the private key.
What most of these library tripped over in that case, is the treatment of the plaintext in a side channel-safe way after the private key operation. For instance, just the simple conversion of an integer to a byte string can be targeted.
Then you'll have non-erased secrets in memory.
Now for RSA specifically, you'll likely have one or more issues linked to: - padding oracles - falling to fake primes due to using non-hardened Miller-Rabin vs Baillie - Confusion on the various PKCS specs - Bleichenbeicher attacks - anything in the Wycheproof repo or cryptofuzz
Certain primes are easier to factor which can weaken your encryption. That was the biggest one when we where taught RSA.
let's p be a prime.
return the set {1,p}.