Yes, there is definitely tile that looks like wood, and tile conducts the heat so much better. Here are a few photos of porcelain wood-tile products:

If you really want wood over your ambient heating system most people suggest Quarter Sawn White Oak flooring as the solution. That factor has played into its surge in popularity especially in the North East.

That's so interesting, Eutree. That's what we used for an addition we built with ambient heating, but only because it matched the original white oak flooring in the kitchen the addition connects to. We just assumed we were going against recommendations in the service of aesthetic continuity. I didn't realize that some hardwoods conduct heat better and warp less than others. Is this correct?

I am currently employed as a Flooring Consultant with a large chain and we get this question all the time !

Every tile company is now coming out with lot of wood tile look tiles THEY LOOK AMAZING INSTALLED !! but not not an inexpensive option

Some things to be aware of when doing tile that looks like wood, typically all boards are the same size approx 6x24 for the ones that I have seen. Most have to be grouted so if you are not fond of grout lines in your wood its not a great option. ( best look Centura Tile they are the distributor and based in Ontario, Canada )
Depending on the type of heat system ( water or electric ) you can use some laminates ( best look are from Armstrong and Torly's )

Another great option is the Sheet FiberGlass glue down or Luxury Vinyl Tile with a Click System ( not glue down ) supplier for both is IVC and they have amazing wood look with amazing durability and HD graphics.

As the others have said, some woods do work again all depending on the type of in floor heating system you have, and the amount of relative humidity you have to deal with. If your hardwood is acclimatized to the temperate that it will be that will decrease the amount of gaping , warping and cupping.

""I didn't realize that some hardwoods conduct heat better and warp less than others. Is this correct?"" @ feeny . Yes this is correct. Traditionally Quarter Sawn lumber will be about 50% more stable than a Flat/Plain sawn product. We mill a good majority of our flooring all the way from log form so that gives us a lot of perspective. :-)