An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This includes two important cases: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore the answer includes Hecke L-functions and L-functions of modular forms.

I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing, and I don't think much is known in general about the modularity for higher $n$. $\mathrm{GL}(2)\times \mathrm{GL}(3)$ alredy seems to be open, see [this answer](http://mathoverflow.net/a/194778/43108) by Paul Garrett.