Does being a degree 2 cover of a projective space impose restrictions on the fundamental groups of nonsingular complex projective varieties? For curves it does not.

2$\begingroup$ A (smooth) double cover of $\Bbb{P}^n$ is simply connected for $n\geq 2$. This follows from the FultonHansen theorem, though this case was certainly known much before. $\endgroup$– abxSep 18 at 9:14
Here is a general statement.
The main theorem of M. Cornalba, Una osservazione sulla topologia dei rivestimenti ciclici di varietà algebriche, Boll. UMI (5) 18A (1981), 323328. Is:
Theorem (Cornalba): Suppose that $f: X \rightarrow Y$ is a branched cover of smooth projective $n$folds, along an ample divisor $D \subset Y$ Then, the following holds
for each $1 \leq i \leq n1$, $f_{*}: \pi_{i}(X) \rightarrow \pi_{i}(Y)$ is an isomorphism.
$f_{*}: \pi_{n}(X) \rightarrow \pi_{n}(Y)$ is surjective.
Note the similarity to the to the Lefshetz hyperplane theorem. Cornalba states in the introduction that this is an analogue of the Lefshetz hyperplane theorem,