The two suggestions so far (too fast for tires; causing heat build up) are both related to what you are asking about.
Each tire compound has an optimum running temperature for best grip and wear. If you go over that temperature, you tend to get less grip and increased wear. This is referred to as "overdriving" your tires.
It really isn't the driver's fault if the tires can't handle the heat created by the car. If we all put snow tires on the NSX and then do some laps, we'd exceed the optimum temperature designed for the tire quickly and then get increased wear and even correspondingly less grip.
The tires need to be appropriate for the use of the specific vehicle. NASCAR gets specific tires for each track. Current F1 regulations provide usually a harder and a softer tire for each track. The whole package needs to work together.
However, driving technique is a factor. As the slip angle gets too great for any particular tire, heat goes way up and grip starts going down. If the driver keeps driving with great slip angles (big understeer or oversteer or even 4 wheel drifts), the increased heat causes more wear and less and less grip. Driver's who have no way to reduce the maximum slip angle may continue in this driving style and may "overdrive" the tires to the point that lap times come down and the tires wear themselves away.
One way to check if you are too hot is with a pyrometer. Use 180 degrees F as a ballpark target temperature. If you are over this, you are probably causing some increased wear and maybe some decrease in grip. Check by coming in from a hot lap directly to the pits and have an assistant look for a maximum temperature across the width of each of the 4 tires.
A stock NSX is fairly well balanced. When the slip angle gets too great, the tires really lose grip and the driver really notices the slower turn speed. NSXes don't have chronic understeer nor oversteer so, in my opinion, standard R compound street tires on and NSX at the track will not lead to a driver tending to overdrive the tires. The stock NSX/R tire combo is good enough that this shouldn't be a problem.
Compare or contrast this to, say, a 1993 Mustang or equivalent. The thing plows (understeers) like a pig and there is no way to drive around it so driver's tend to drive way over optimum slip angles, causing increased heat and premature wear.
"Overdriving" comes from an improperly setup car - one with too much understeer or oversteer. If your NSX has one of those, then it isn't set up correctly - the good news is that it can be fixed. Fixing some cars is a major change to the suspension so be glad you have an NSX.
If you want to analyze this further, it would help if you describe how the car felt as the tires went away. Did you notice more problems at turn in where the car won't turn in as well towards the apex? Or, did you notice the rear end coming around in turns when you didn't want it to? This will likely point to which pair of tires is getting overheated.
I hope this helps.
