Honcho's answer is correct / incorrect depending on the circumstances of your problem. If the ignition key is switched to the run position and the radiator fan is running with a cold engine, then he is likely correct that the Fan control unit is the problem; but, a failed temperature sensor could also be the problem. Be aware that the fan unit suffers the same problems as the other NSX electronics (solder cracks and I believe capacitor issues). If you diagnose the fan unit being the problem then pull it apart and examine it and it may be repairable.
Because you said that the fan is running until the battery is dead, I take it that the radiator fan is running even when the ignition is switched off. If so, then I respectfully submit that Honcho might be incorrect. This is the wiring diagram for the fan control
The radiator fan itself is supplied off of un switched power. However, both the fan control unit and the fan high and low relays are supplied from switched power. Unless your vehicle has had botched wiring surgery, there is no way that the fan control unit can power up either of the two relays to switch the fan on when the ignition switch is off. If your fan is running when the ignition is switched off, then you have one of three problems:
1) you have a short to ground somewhere on the wht/blu wire coming off the fan motor. This could include a melted fan relay socket which is allowing a ground short
2) the high or low fan relay internal contacts have welded themselves shut
3) the fan relay overheated and has melted the socket which is allowing a short to ground on the wht/blu wire.
You said you checked the relays. How did you check the relays and did you check the right ones? From year to year Honda moved the relays around. If you used a 1991 service manual to determine relay location it shows both fan relays are in relay box A. As of 1997, the high relay switched positions with the horn relay in box A and the low relay moved to box B.