They only influence aesthetics, really.
Kidding. The densities of wood can influence sound, though as per Zach Merbach of ZMF the differences between varying woods on the same models on his headphones are largely negligible; a large part of what contributes to the final sound is design. My Klipsch HP-3s have solid ebony cups but are sonically identical to the Oak and Walnut variants due to the inside of the cups being treated with a lacquer that apparently keeps the sound standard— due to the treating of interior surfaces I wonder whether substituting the wood for anything similarly dense without the acoustic characteristics of wood might affect the sound.
The TH-X00 is a better example of how wood can influence headphone presentation, as the internal surfaces are untreated. The Ebony variant is reasonably linear (albeit downward sloping with a slight tick upwards in the mid-treble region), while the Purplehearts have more pronounced low midrange and low treble dips, and more interesting resonance characteristics that convey a pleasurable colouration to the sound besides (Mahogany splits the difference, but I think is closer to the EB overall?). It’s not just a matter of FR balance but also of distortion characteristics/internal reflections/loads of other things. Species of wood aside the shape of the cups, internal volume, thickness, and heck maybe even natural variation between samples of wood can all contribute to the final sound. I’ve always wondered what putting a modern TH900 driver into a Sony MDR-R10 with dead diaphragms (primitive biocellulose tech ftw) might do.
I don’t really play any instruments but it may be worth looking into tone woods and how they influence acoustics. I’ve some friends who play guitar/violin and they could go on for ages about how wood matters when designing and selecting an instrument. This all is a massive oversimplification, but I’m more a subjective guy than a technical one to begin with, haha. I personally like how well-done wooden headphones sound— they don’t necessarily have the clinical precision of Sennheiser’s hi-fi products, or HiFiMan’s for that matter, but those subtle tells you get when listening to wooden headphones can be quite pleasing. I’m excited to eventually get ears on an Auteur or Aeolus, they seem up my alley.
Challenges, well I imagine making consistent products could be a massive pain in the bunghole given how wood is a natural product and inherently variable. I also know of some species of wood that are so hard as to damage tools when working with them. Softer woods might be easy to play with on a lathe but are consequently less durable. Also, living in a tropical climate I can attest to how poorly-finished wood is susceptible to swelling and disintegrating (this is hyperbole, just about) over time.
I remember reading random articles online about how material choice affect performance in loudspeaker designs. Not exactly headphones, but some of the same principles will apply I daresay. Trying to track those down now.