I think hot-stamps benefit from oil more than inlaid chips, because there's more exposed "clay." It's the "clay," not the inlay, that takes on the oil into pores and loses a bit of the whitish, chalky sheen caused by light being scattered.
Hot stamps aren't hurt by the oil - the big risk is harsh cleaners. Don't use bleach or oxidizing cleaners on inlays.
Also, in case anyone cares, the oil doesn't actually "dry." There's no water in the oil, and oils don't evaporate.
Mineral oil never really dries; it seeps into the pores and fills gaps, but stays "oily." That's why you need to only use a little - over-oiling will just leave a film of oil on your chips, which may also lead them to pick up dirt easily. You can just wipe them down with rags after oiling, and a small bit of excess oil will "soak in" gradually. Over time, it will transfer (into people's fingers, the felt, or deeper into the chip), but it doesn't actually dry.
Linseed or canola oil, on the other hand, will "cure." They will actually turn hard. Again, technically, they don't "dry" because there's no evaporation involved - instead, in the presence of air, the oil will polymerize. What happens is that chains of oils are like threads, and where they are touching, they'll form cross-links. The cross-links require oxygen to form. So an open cup of oil can't polymerize, except for the very top layer, which can form a "skin." When you oil something with a "drying oil" like linseed, the thin film of oil on the surface will slowly polymerize in the presence of air. Once it polymerizes, it forms a dry, tough, rubbery coating on the surface. It's embedded into the surface of the chip, and you can't even see it, other than the same "oiling" effect you get from mineral oil. However, several coats of that kind of oil, each coat a day apart, will put a deep glass-like sheen onto a chip or a woodworking project. That's not what you want, and it would be hard to get off.
I've been happy with the results from mineral oil, and even from canola oil (sold in the grocery as "vegetable" or "salad" oil.) I think the canola-oiled chips stay clean better, but mineral oil is easier in the sense that you don't need to leave them out in the air to cure after oiling. Also, with mineral oil, you never have to worry about it building up - if there's too much, you just wipe them down. Layers of a "drying oil," on the other hand, might build up over time in the recesses and the mold marks, and will eventually just look like gunk. So if you plan to oil your chips often, don't use a drying oil.
When you wash your chips, the detergent will break down and remove oils. So the chips will have a 'dryer' look; depending on microscopic surface texture, they may even look white and chalky because of the way light refracts (think frosted glass.) Oiling after washing is, to me, a necessity.
So if you're the kind to wash and re-oil your chips often, you may lean towards mineral oil, to avoid build-up.
For a set that that won't get that kind of treatment, canola may actually work better. I feel like my Paulson Fun-Nite stayed clean longer after a canola oil treatment than after mineral oil.It's hard to tell, though, because I was playing them more after the mineral oil, and with different people.
Good luck with your chips!