When defining Weil divisors, Harsthorne considers only schemes which are integral, noetherian, regular in codimension 1 and separated. I am wondering if one can drop the condition that the scheme is separated, as I don't see how this plays a role in the usual definitions and properties of divisors. Is there anything that goes terribly wrong? :)
(I am aware that the other conditions are not all relevant for every definition).
Thank you!
Added: I noticed that your question was also about the usual definitions of Weil divisors, thus I want to add the following comment:
When $X$ is nonseparated, a discrete valuation on $k(X)$ no longer determines a unique prime divisor on $X$ (Hartshorne Ex.II.4.5). In particular for the line with infinitely many origins, the principal divisor corresponding to $1/t \in k(X) = k(t)$ is an infinite sum. Stacks Project gets around this by defining Weil divisors to be locally finite sums - when $X$ is quasi-compact this recovers the original definition.
New answer:
I flipped through Hartshorne section II.6, and nothing requires separatedness there that I could see. The following doesn't precisely address the question, but I added it because it helped me lose interest in the question :).
Fitting with our intuition for the line with doubled origin, Stacks Project Lemma 27.29.3 says that if $X$ is quasicompact then we can find a dense open subscheme $U \subseteq X$ which is separated. Then by Hartshorne Prop. 6.5 there is a surjection $Cl(X) \to Cl(U)$, which is an isomorphism when codim$_XZ > 1$. So, in many examples the class group is the same or not much different than that of something separated. Lemma 27.29.3 is also still satisfied for the easiest of non-quasicompact examples, like affine space with infinitely many origins.
If you are hunting for pathologies, maybe you should look for something that fails 27.29.3 then. Still though, I'm not sure what properties you would look for to fail since I think everything in Hartshorne is still true.
Upon closer inspection, the paper I linked to below has more to do with nonreducedness and is about Cartier divisors rather than Weil divisors. My apologies for that. (In particular, the paper is about when Hartshorne Prop. 6.15 fails, and 2.2 is a non-reduced scheme example, which so happens to also be nonseparated.)
Original answer:
I don't know something specific that goes wrong, so maybe this will not answer your question - my apologies if none of this is news to you.
I want to point out that at The Stacks Project Weil divisors are defined for $X$ a locally Noetherian integral scheme. I think these assumptions are just enough to make principal divisors well-defined; I recommend looking through the Stacks Project chapter for more information.
Also, when someone is wondering "why did Hartshorne make the assumption _____" (which happens often), they should go to the Stacks Project to see the most general thing possible.
Update:
Have you seen this paper of Schroer? Example (2.2) might be what you are looking for.