I've been thinking about this, and I want to record a few thoughts. Let $k$ be a field of characteristic $p$, let $X$ be a smooth $n$-dimensional variety, let $D$ be a Cartier divisor and let $U = X \setminus D$.
We cannot hope to have a natural map from $H_{DR}^n(U)$ to $H^{n-1}_{DR}(D)$ which looks anything like the residue map (also known as the Gysin map). Take $p$ odd. Take $X$ to be the affine plane with coordinates $(x,y)$, and let $D$ be $\{ y=0 \}$.
Let $\alpha$ be the $2$-form $x^{2p-1} y^{-p-1} dx \wedge dy$ and consider the automorphism $\phi(x,y) = (x+y,y)$ of $X$. This preserves the divisor $D$ and acts trivially on $D$, so $\alpha$ and $\phi^{\ast}(\alpha)$ should have the same residue. In other words, $\phi^{\ast} \alpha - \alpha$ should have residue $0$. Now, $\phi^{\ast} \alpha - \alpha = \sum_{j=0}^{2p-2} \binom{2p-1}{j} x^j y^{p-2-j} dx \wedge dy$. The term $x^j y^{p-2-j} dx \wedge dy$ is exact whenever $j \not \equiv -1 \bmod p$, so (if we want a map that factors through $H^n(U)$), the residue of $\binom{2p-1}{p-1} x^{p-1} y^{-1} dx \wedge dy$ should be $0$. Also, $\binom{2p-1}{p-1} \equiv 2 \bmod p$ by Lucas' theorem. So the residue of $x^{p-1} y^{-1} dx \wedge dy$ should be $0$ in $H_{DR}^1(D)$. Note that $x^{p-1} dx$ is not zero in $H_{DR}^1(D)$, so we have to do something non-obvious.
There is a very deep thing to do. We can lift $X$, $D$ and $U$ up to flat schemes over some dvr of mixed characteristic (for example, if $k = \mathbb{F}_p$, we could take $p$-adic lifts) and take the de Rham cohomology of these lifts. There is a ton of very hard literature on this sort of idea, starting with the research of Monsky and Washnitzer. Indeed, there is a Gysin sequence in Monsky-Washnitzer cohomology: See
Monsky, P., Formal cohomology. II: The cohomology sequence of a pair, Ann. Math. (2) 88, 218-238 (1968). ZBL0162.52601.
I don't feel confident to summarize this paper.
I went looking for something more elementary to do and came up with an interesting idea: Although $x^{p-1} dx$ isn't exact, it is in a sense "almost exact". The exact forms are the kernel of the Cartier operator, and $x^{p-1} dx$ is in the kernel of the square of the Cartier operator. Define $EH^n$ to be $n$-forms modulo forms which are killed by some power of the Cartier operator. (This is a definition for top dimensional forms only; see my recent question for what I think the more general definition should be.) I think I can build a Gysin map $EH^n(X) \to EH^{n-1}(D)$. But I'm going to wait a bit to see if someone answers my other question before I write more.