Let $M$ be a model of ZFC and $M\models \varphi$ such that $\varphi$ is a $\Sigma_2$ sentence in the language of set theory. Let $M[G]$ some forcing extension of $M$.
Is $M[G] \models \varphi$? What about $\Sigma_3$ sentences? $\Sigma_2^1$ sentences?
First we should clarify what absoluteness between $M \subseteq N$ mean. I will assume that $M$ and $N$ are transitive sets. $\varphi$ is absolute between $M$ and $N$ means that $M \models \varphi$ if and only if $N \models \varphi$. (Note that if and only if.)
From your question, if $M \models \varphi$, then $N \models \varphi$ is just upward absoluteness or persistence.
So note that $\Sigma_1$ sentence are not even absolute. Consider $\varphi(x,y)$ defined by
$$(\exists f)(f \text{ is a bijection between } x \text{ and } y)$$
You can check that this is $\Sigma_1$. Now let $\mathbb{P} = \text{Coll}(\omega, \omega_1^M)$. Let $G$ be a generic for this forcing. Then $M[G] \models \varphi(\omega, \omega_1^M)$; however, $M \not\models \varphi(\omega, \omega_1^M)$. This $\Sigma_1$ statement is not absoluteness since it is not downward absolute.
As all $\Sigma_1$ formulas are $\Sigma_2$, $\Pi_2$, $\Sigma_3$, etc., these formulas are in general not absolute.
Now addressing your question which seems to be about upward absoluteness. First note that the example above will not work since by basic model theory, all $\Sigma_1$ formulas are upward absolute.
Now consider the formula $\varphi(x,y)$ which asserts $y$ is the power set of $x$. This is:
$$(\forall z)(z \subseteq x \Leftrightarrow z \in y)$$
which is $\Pi_1$. Let $\mathbb{C}$ denote Cohen forcing. Let $G$ be a generic filter for Cohen forcing. Let $x_G$ be the generic subset of $\omega$ added by generic filter. Then $M \models \varphi(\omega, \mathcal{P}(\omega)^M)$. However, $M[G] \not\models \varphi(\omega, \mathcal{P}(\omega)^M)$ since in $M[G]$, $x_G \subseteq \omega$ but $x_G \notin \mathcal{P}(\omega)^M$. So this $\varphi$ is not upward absolute.
Again, since $\Pi_1$ sentences are $\Sigma_2$, $\Sigma_3$, etc., this is an example for what you were asking for.
The $\Sigma_n^0$ and $\Sigma_n^1$ hierarchy are different. In the above, the quantification is over all sets. Roughly, $\Sigma_n^0$ is quantification over natural numbers and $\Sigma_n^1$ is quantification over reals. (Look up the definition of the projective hierarchy for more precise definitions.)
By absoluteness, I mean the "if and only if" sense in the first paragraph.
In $\text{ZFC}$, $\Sigma_1^1$ formulas are absolute between transitive models of $\mathsf{ZFC}$ (in particular generic extensions). This is Mostowski Absoluteness.
Even more general in $\mathsf{ZFC}$, all $\Sigma_2^1$ formulas are absolute between transitives model of $\mathsf{ZFC}$. This is Shoenfield absoluteness.
Immediately from Shoenfield absoluteness, $\Sigma_3^1$ sentences are upward absolute and $\Pi_3^1$ are downward absolute.
These are limits of projective absoluteness in $\mathsf{ZFC}$. There are models where $\Sigma_3^1$ sentences are false between some model of set theory and its constructible universe $L$.
However, in a universe with sufficient large cardinals assumptions or determinacy assumption, one can have $\Sigma_n^1$ absoluteness for any $n$ or all $n$.