Generally I don't bother, but if the jig is a bit more complex than just a couple of bits of MDF screwed together, I may well give it a coat or two of clear lacquer, especially any nice hardwood knobs, stops and the like. Sliding parts usually get waxed.
For something as substantial as my UTTJ it gets finished as I would any other piece.
I finish mine but only because I hate wasting finish and you're not supposed to put finish back in the tin because it's got bits in once you've been dibbing into it with your brush or whatever
so when I'm doing some finishing I get all my jigs out which may or may not have already been partially finished and then go along them using up the left over finish until there isn't any left over finish
not sure a table saw push stick actualy needs 20 coats of various finishes and stains but hey ho!
I've never done it, though I like the idea of cleaning your brushes onto them. My jigs are mostly plywood, pine, masonite, and East Asian palletwood, and as long as they don't get wet should last indefinitely.
I place finish on my MDF/HDF jigs that are used on CNC and are to use vacumn facility. MDF/HDF is porous and you can leak valuable sucking power, I have seen PVA used as an alternative. I much prefer good quality Brirch plywood for jigs now as they are sturdier and do not have leak issues.
Most of my work jigs are made of either Keruing offcuts which are resistant to just about any attempt to finish them (and in any case will periodically exude resin when conditions change) or from welded/riveted steel bar-stock, the latter tend to be finished with boiled linseed oil, which if built up and allowed to form a "dried" layer helps keep the worst of the rust off.