Honestly, I really doubt anybody is going to know the answer to this kind of question. I'm not about to experiment either! I recommend you try it out on the neck pocket (hidden) area, let it dry and see how it feels.
Dying it won't affect the tone in any way. I personally wouldn't try dying it, but if you're determined I think the wood is rather "oily" so you'll probably need to use a black universal dye concentrate diluted in lacquer thinner; experiment on the back of the heel first.