Done all three many times.
Metal - Expect the guitar's tone to get a lot brighter, though by how much depends on the type of metal used and the mass of the pickguard. Also you need to be careful to isolate all the electronics from the guard, or don't play anywhere with lots of neon lights and dodgy wiring. It's hard to screw up the construction of metal guards and I wouldn't worry about buying them off Etsy, eBay, etc.
Wood - No discernible change to tone. Make sure it's either multi-ply or a really dense wood like rock maple or walnut, and no matter the wood species be sure get some kind of finish on it, even if it's just a basic furniture wax. Wood's a pain for pickguards if you want lots of small switches drilled or multiple pickups close together, like an 'Elite' config Strat for example, unless you make the guard extra-thick; thin wood guards tend to be on the brittle side. For something like a Tele or a LP Jr it works fine. Poor craftsmanship with wood can obviously be a problem no matter what the object is meant to be, so I'd only buy a wood guard if you can be really sure the supplier knows what they're doing. I've gotten them custom cut by a local furniture repair guy; I don't think I'd trust any randoms online.
Leather -
Very slightly dulls the guitar's tone compared to having a slab of plastic there, unless you put it on a stiff secondary guard of some kind (either wood or a plastic), in which case it has no effect on tone. Works totally fine, you just have to be sure it's all locked down tight and all the edges are sealed nicely, 'cause you don't want any raw leather getting grotty (assuming it's real leather) or loose bits flapping about. Any good leatherworker will be able to deliver a guard that is just as good a fit as any plastic one and all properly finished, but as is the case with guitar straps, there are a lot of sloppy leather products out there so I wouldn't buy from any old random or from any faceless mass-production factory using battery-farmed leather. Make sure to buy from an experienced leatherworker, someone who makes it their whole business and only uses quality materials. Personally I only get leather pickguards from
Beth Freeman, who is also who the Fender Custom Shop use. She's not cheap (£200+ per guard) but cheap leather is not something you want to mess with.