I have never had a MIL warning on my NSX (knock on wood!) so I have not experienced your problem. That said, does the code reader not detect a code or is the code reader unable to read anything from the ECU (you don't need to have a MIL warning light on to be able to read data from the ECU)? If they can't read anything from the ECU, the code reader may not be communicating with the ECU which could be as simple as a faulty connection at the OBDII port or a broken wire between the port and the ECU. You need to determine whether there was a communication problem or whether the reader is communicating with the ECU; but, is just not detecting a code.
The MIL lamp gets a constant 12 v supply from the dash and is lit up by getting a ground connection from the ECU when an error code is generated. If the wire connecting the MIL lamp to the ECU is grounding out somewhere on the chassis, the MIL lamp will light up even though there is no code in the ECU. If the code reader can communicate with the ECU, then check for a bad connection between the MIL lamp and the ECU.
16 volts sounds like the voltage regulator is buggered and is applying full voltage to the alternator field winding. Around 16 volts is typical for the open circuit saturation voltage on automotive alternators. Confirm that the voltage is flakey (and that it is not the dash voltmeter that has gone south) by using a multimeter to measure the voltage to ground. The spot in the engine fuse box used for jump starting would be a good test point. If the system is actually running at 16 volts, you could try retrofitting a new regulator / brush set on your alternator. I have never had much luck with that. I suggest new alternator or rebuild if you have a good source.
Really low voltage can trigger the MIL light because it causes the ECU to interpret that the sensing devices are operating out of range. In such case you would have a code stored in the ECU. I have never heard of that problem being caused by high voltage in part because I would expect the internal voltage regulators in the ECU are capable of managing a 16 volt source voltage just fine. However, 16 volts will lead to rapid aging of your light bulbs and eventually destroy your battery, so get the problem fixed.