Assume, in $V$, that we have some forcing $P$, a $P$-name $\tau$ and some sentence in the forcing language $\varphi(\tau)$ (it may include other names but we focus on $\tau$). Now we take some countable elementary submodel $M$ such that $P,\tau$, the maximal element of $P$ and every other name mentioned in $\varphi$ are in $M$.
Assume that $P\Vdash \varphi(\tau)$. So in $M$, $P^M=M\cap P$ is also a poset with the same maximal element, $\tau^M$ is also a $P^M$ name, and by the definability of the forcing relation we can relativise $[P \Vdash \varphi(\tau)]^M$ so in $M$, $P^M\Vdash \varphi^M(\tau^M)$. (Was that right?)
Now we go back to $V$ and look at $P\cap M$. If I get it right, it is still a poset, and $\tau^M$ is still a name for it. Now I want to force with it over $V$, say with a generic $G\subset P \cap M$. Will I get that $V[G]\vDash \varphi((\tau^M)_G)$?
I believe the answer is "yes" from the elementarity and the fact that the forcing relation is definable, but there might be something I'm missing.
Also - if it matters we can assume (for the cases that interest me) that also $\tau \subset M$ - does it matter?
I think the answer is no.
Starting with a ground model of $V=L$, let $\mathbb{P}$ be the Levy collapse of $\omega_1$ to $\omega$, and let $\varphi$ be the sentence "$\omega_1^L$ is countable." (Note that there is no $\tau$ here, since $\omega_1^L$ is definable. But if we like we can take $\tau$ to be the canonical name for $\omega_1^L$.) The point is that from $V$'s point of view, $\mathbb{P}\cap M$ is isomorphic to Cohen forcing, since $M$ is countable. So forcing with $\mathbb{P}\cap M$ over $V$ won't collapse $\omega_1^L$.
Am I misunderstanding your question?