Fundamental group of an open subscheme of a normal scheme

Let me expand my comment into an answer.

Take as $X$ the cone of vertex $v$ over an elliptic curve $E$. Then $X$ is simply connected (this is a general property of projective cones). However, $U = X-\{v\}$ is not simply connected: in fact, the projection $\pi \colon U \to E$ onto the basis gives $X$ the structure of a topological fibration with fiber homeomorphic to $\mathbb{C}$, so the corresponding long exact sequence of homotopy groups yields $$\pi_1(U) = \pi_1(E) = \mathbb{Z} \oplus \mathbb{Z}.$$

In fact, quite the opposite tends to be true. Mumford [1] showed that for $(X,0)$ the germ of a normal surface singularity (over $\mathbf{C}$), $U=X\setminus 0$, one has $\pi_1(U)=\{1\}$ if and only if $X$ is smooth. At the same time, $\pi_1(X) = \{1\}$ since $0\to X$ is a homotopy equivalence.

If $X$ is smooth, this is true (the etale variant is called "Zariski-Nagata purity").

EDIT. To address Francesco's comment: of course the example is not projective. The easiest projective example was given by Francesco in his comment: $X$ is the (projective) cone over an elliptic curve $E$ and $U$ the complement of the vertex. Then $\pi_1(U)= \pi_1(E) = \mathbb{Z}^2$ and $\pi_1(X) = \{1\}$.

[1] Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math., Inst. Hautes Étud. Sci. 9, 5-22 (1961). ZBL0108.16801.

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.