I'm very new to the subject of forcing. I undertand that if we have a countable transitive model $M$ of $\sf ZF$, and if we have an element $\mathbb P$ of $M$ that is partially ordered by relation $\leq$ in atomless manner, with a maximal, and if $G$ is a generic filter on $\mathbb P$, then the set $M[G]$ of all $G$-values of $\mathbb P$-names in $M$ is also a model of $\sf ZF$ that has $G$ among its elements, and actually its the minimal model of $\sf ZF$ that has $M$ as a subset of it and have $G$ among its elements. For explanation of terminology see the wikipedia article on mathematical forcing.
My question suppose that $M$ is a model of some extension $\sf T$ of $\sf ZF$, and by extension I mean a theory that adds primitives to the language of $\sf ZF$ and of course consistently adds axioms about them on top of axioms of $\sf ZF$. Would it always be the case that $M[G]$ be a model of $\sf T$ also?
Certainly not, even if we don't change the language at all! In fact the whole utility of forcing (as Daniel Schepler points out below) is based on the fact that many things aren't "forcing-stable." For instance, $\mathsf{CH}$ is not forcing-stable (this was Cohen's first application).
In general, in fact, I would say that it is rather surprising when a statement is forcing-stable. The axiom of choice is forcing stable, as are the $\mathsf{ZF}$-axioms themselves, but - for example - the axiom of determinacy is extremely forcing-unstable, in the sense that a wide variety of forcings are guaranteed to break $\mathsf{AD}$ (see e.g. Chan/Jackson or Ikegami/Trang).
As far as I know, there is still no comprehensive rule of thumb for determining whether a given statement is preserved by (a certain type of) forcing. Certainly some important special cases are well-understood - e.g. Shoenfield absoluteness, or that countably closed forcings don't add reals means that they also don't change the theory of $L(\mathbb{R})$, or that if there is a proper class of Woodin cardinals then no (set) forcing whatsoever can change the theory of $L(\mathbb{R})$ - but overall there's still a lot of interesting stuff to be done here (to put it mildly!).
Incidentally, and admittedly contra my own stylistic choices above, in my opinion forcing-stability should properly be thought of as having both sides variable - that is, we should say "$S$ is forcing-stable over $T$" rather than "$S$ is forcing-stable." This isn't just aesthetic. For example, the fact that forcing preserves admissibility (= "$\mathsf{KP}$ is forcing-stable over $\mathsf{KP}$") is quite useful independently of the forcing-related behavior of full $\mathsf{ZF}$.