Let $X$ be a smooth projective algebraic variety over a field of characteristic zero and let $D$ be a simple normal crossing divisor on $D$. Put
$j: U \hookrightarrow X$
for the inclusion of the complement $U=X-D$ on $X$.
Are the functors $j_\ast$ and $j_!$ exact?
I think the answer is yes.
I would be very grateful if someone could provide me with a proof.
I assume you're asking about quasicoherent sheaves.
Since the inclusion of $U$ into $X$ is an open immersion, $j_!$ is exact. You can find this in Tag 03DJ.
Since $j$ is an affine morphism, $j_*$ is exact (EGA 2 Corollary 5.2.2). $j$ is affine, because the affine property of morphisms is étale local on the target, and the inclusion of the complement of a principal closed subscheme (such as the locus defined by $x_1 x_2 \cdots x_r = 0$ in affine space) is affine. See e.g., Tag 07ZT
For a general open immersion $j: U \hookrightarrow X$, $j_!$ is exact (it is left exact and left adjoint to $j^*$), but I would guess that $j_*$ is (in general) only left exact (it is right adjoint to $j^*$).
