The obvious sufficient condition for $v\mapsto f(v)$ to map $H^1(\Omega)$ to itself is $f\in W^{1,\infty}(\mathbb R)$, i.e. $f$ globally Lipschitz and bounded.

This condition is also necessary for $N\ge2$, I think (but don't ask me for a reference, that's folk wisdom, although it must be proven somewhere).

The obvious sufficient condition for $v\mapsto f(v)$ to map $H^1(\Omega)$ to itself ($\Omega$ bounded) is $f$ globally Lipschitz, i.e. $f'\in L^\infty(\mathbb R)$. Add $f(0)=0$ for general $\Omega$.

This condition is also necessary for $N\ge2$, I think (but don't ask me for a reference, that's folk wisdom, although it must be proven somewhere).