Tone woods obviously affect acoustic/classical instruments, and have been a point of intense debate with electric guitars. Some say that wood is important, while others say wood is unimportant. I can personally report that when you are playing an electric guitar, the differences in body/neck feel and tone can be quite obvious. However, these may not remain audible after processing, mixing, and recording. Electrical processing adds a huge set of additional variables.
With headphones, again you must consider whether the wood is integral to audio reproduction or just kind of sitting there on the side. You might feel the differences between woods by touching the cups more than you hear them. Furthermore, one should consider whether the frequencies present in musical content correspond to the resonant range of the wooden cups. The composition of the drivers (as plastic, paper, metal, etc.) will overshadow the passive components.
One of the better guitar comparisons I’ve seen involved creating three nearly identical guitars with different woods from wooden planks. I hear the differences, but part of this results from using thin wooden planks whereby some flex more than others:
For additional guitar analyses, also see this, this, and this.