Currently using a printer with dye inks but I am considering replacing it with a pigment based one. Are papers always compatible with both? Or do some work with one or the other? How can one tell if some paper is suitable for pigment inks?
In short, no, they are not always compatible.
I have had photo papers (Kodak I think) that actually rejected Epson pigment (Ultrachrome) inks (it sat on the surface and refused to dry properly.
I have also seen certain dye inks leech under the surface of glossy photo papers leaving (in one case) a magenta halo.
Pigment inks tend to leave a matte finish on glossy papers, but work well on matte papers as they dont soak in and bleed (they dry in the surface), whereas dye inks tend to leave a super glossy finish on glossy papers, but a flat, dull finish on matte papers.
The only way to know for sure is to test, luckily most manufacturers do tester packs so you dont have to potentially waste £20, £30+ on a pack.
If it's glossy or coated, read the label. They say "for all inkjet and laser printers" or whatnot. Paper made before that printer technology won't necessarily work, but new paper will be labeled as to its compatibility. Papers made by a paper company, not branded as the printer company, can be expected to be versitile.
I had trouble with the epson dye not soaking into the Kodak glossy paper, but a year later never had any problems with paper compatibility. The problem was the dye not soaking into the paper fibers; the opposite of another poster. So I think the issue is with the solvent not with what's in it, and you can't tell from "dye"/"pigment" how it behaves.
Your common consumer won't be able to handle that. So except for specilty papers meant for specific printers, you can assume that paper so-labeled (read the label) will work on any ink-jet print engine that's been out for a year before the paper was made. And I don't expect any surprises from newer ones: subtle quality issues, not outright incompatibility.