The fact is, if an amp has:
1) inaudibly low noise
2) inaudibly low distortion at the required power and current level
3) output impedance <1/10 of headphone impedance at all frequencies
4) sufficient voltage swing to drive the headphones to an acceptable level
5) inaudibly low channel imbalance
6) a flat frequency response
7) Little or no overshoot or ringing regardless of load
8) Decent phase response
the amp will sound just like every other amp that meets these criteria.
It isn't terribly hard to meet these criteria with modern electronics either, and any competently designed amp should succeed at this. As such, any competently designed amp with sufficient power and voltage swing (and this just means enough to get the headphones fairly loud) will drive the HD800 perfectly. Note that it only takes ~160mW at 8Vrms to get to 120dB with the HD800, so the limiting factor for nearly any amp out there is going to be voltage swing, not current capability (and even the amount of voltage required to get them ridiculously loud is well within the capability of most reasonable amps). The HD800 is actually a fairly easy load, from the amp's perspective - high impedance, low current, and not hugely insensitive. Something like an LCD2 is much more challenging to the amp than the HD800 ever could be.
