No, in fact, the integral has no reason to even converge.

Take $X=\mathbb{CP}^1$, D a point, and so $X-D=\mathbb{C}$. Then $\alpha(z):=|z|^2$ is a smooth function on $\mathbb{C}$, whose laplacian is a constant.

EDIT: I am interpreting the question by assuming that $\alpha$ is a smooth function on $X-D$, but the last phrase of the question suggests that this interpretation is maybe incorrect (if $\alpha$ is a smooth function on $X$, then the integrals on $X$ or $X-D$ are obviously the same because $D$ is a subset of easure zero).

No, in fact, the integral has no reason to even converge.

Take $X=\mathbb{CP}^1$, D a point, and so $X-D=\mathbb{C}$. Then $\alpha(z):=|z|^2$ is a smooth function on $\mathbb{C}$, whose laplacian is a constant.

EDIT: I am interpreting the question by assuming that $\alpha$ is a smooth function on $X-D$, but the last phrase of the question suggests that this interpretation is maybe incorrect (if $\alpha$ is a smooth function on $X$, then the integrals on $X$ or $X-D$ are obviously the same because $D$ is a subset of measure zero).