Let $\psi^n_t$ be a sequence of stochastic process that converge for all $t$ to $\psi_t$ a.s., namely $\psi^n_t \to\psi_t$. In the following case, for bringing inside the limit, is it necessary applying some convergence theorem, as for example, the lebesgue's dominated convergence? $$\lim_{n\to \infty}\int_0^t\psi_s^nds$$ and concluding that $$\lim_{n\to \infty}\int_0^t\psi_s^nds \to\int_0^t\psi_sds $$
Which are the condition that has to be satisfied?
That there exists a stochastic process $\varphi$ such that $|\psi^n_s|\leq \varphi_s$ for every $s$ and $\varphi\in L^1([0,T])$ for every event. If you only want the limit to hold almost surely, you can replace all of the above identities appropriately.
This, of course, not necessary, but probably the most general theorem that is generally applicable.
Alternatively, if $\psi^n\to \psi$ in $L^1([0,T]\times \Omega),$ then you get the desired by Fubini's theorem, but this requires additional measurability assumptions.