The "problem" with birdseye maple is that the eyes are denser than even the surrounding (already dense) maple. If you managed to somehow dye it - the eyes would be LIGHTER because they're denser. Not sure if thats the effect you want.
The amber Jack mentions is a dye based toner. You MIGHT be ok with something like a very very thin wash of dye - I mean THIN - like mostly acetone or MEK, and very little dye. The effect is not really to "color" the wood but to give it some pop. Can't say for sure if it will pop like you want it to.
Be sure to thin your Tung oil too - on the first coat for better penetration. Be sure to wipe your tung oil OFF the neck after every coat. You want it to be as thin as possible - just a bare coat. Wiped on, set to set a little, ALL wiped off, except is there in the wood. If you do it that way you'll end up with a gorgeous finish.