Although $X$ and $U$ need not have isomorphic fundamental group (as the accepted answer shows), the induced map from the inclusion $U\hookrightarrow X$ is generally $\pi_1$-surjective.

Precisely, if $X$ is a normal projective variety, and $A\subset X$ is a proper closed subvariety, then the natural homomorphism $\pi_1(X-A) \to \pi_1(X)$ is surjective.

You can find this result in *On the fundamental groups of normal varieties* by Donu Arapura, Alexandru Dimca, Richard Hain.

Interestingly, although natural inclusions only give $\pi_1$-surjections, it turns out that natural projections do give $\pi_1$-isomorphisms.

Precisely, let $G$ be a connected reductive algebraic affine group over an algebraically closed field $k$ (arbitrary characteristic). Assume $G$ is acting on a smooth connected projective variety $M$ (and there is an appropriate ample line bundle $\mathcal{L}$). Then the homomorphism (induced by the GIT projection) of algebraic fundamental groups $\pi_1(M)\to \pi_1(M/\! /_{\mathcal{L}}G)$ is an isomorphism. If $k = \mathbb{C}$, then there is also an isomorphism between the topological fundamental groups.

You can find this result in *Fundamental group of a geometric invariant theoretic quotient* by Indranil Biswas, Amit Hogadi, A. J. Parameswaran.