It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas Weil cohomology theories should take values in characteristic $0$. It turns out that de Rham cohomology picks up torsion from Crystalline cohomology, so the dimensions are actually not the correct ones.

However, even if it's not a Weil cohomology theory, one could still try to prove some of the properties. The one I am interested in is Poincaré duality, including the construction of the cup product pairing, say for smooth proper varieties.

I think I know how to do this, but it is a bit technical. I was wondering if there is a place in the literature where it is carried out. It seems that most sources on algebraic de Rham cohomology immediately assume characteristic $0$, and don't prove the results for characteristic $p$ (presumably because de Rham cohomology is 'not the correct thing to consider').

Edit. Maybe I should have stated more clearly what I already know, and what it is that I am looking for. I am certainly convinced of the truth of the statement. For $X$ projective, there is a nice Čech-theoretic approach, worked out below by David Speyer. In general, there is a dévissage approach where we truncate the de Rham complex and induct on the number of terms. One place where one can read about this is these notes (p. 3-4) by Johan de Jong. In the notes, it is only carried out for $X$ projective, but the same approach should work for $X$ proper. (The notes assume characteristic $0$, but this is not used in the construction.)

However, the reason I posted this reference request is that I want to use the result in a paper. I prefer using peer reviewed sources to informal notes, and this is the type of result that should have been known for more than 30 years. I was hoping someone happens to know where to find it.