What I have till now is that $2016= 2^5\cdot3^2\cdot 7$.
Also, because $m+n+mn=2016$ then $m$ and $n$ must be even. For the rest my idea is to use congruence module $3$, and $7$ to see all cases. Do you have a better idea? Because there are a lot of cases. How would you find the solutions?