While some kids do give up naps early, I have found that many kids still need naps until about age 4 or longer. Many kids that I've cared for go on what I call and "nap strike" at about age 2 /12 to 3. They have figured out that they don't *have* to nap and feel ok some of the time and try to not nap anymore at all. But just as you describe they can only handle it for a couple of days before they're overtired little bears.
I take the control out of it. I tell kids they don't have to sleep, but they do have to have quiet time. If, after an hour, they aren't asleep I let them get up. Many kids can keep themselves up for a half an hour, but not usually for an hour if they are really tired enough to nap. If they don't fall asleep, the hour of resting time is usually enough to recharge them to make it through the rest of the day without melting down at 5pm.
Have him rest at the same time every day (just like nap time), in a quiet, dim room. He has to lay down. No singing or taking with stuffies...Some people let their kids look at books, but I never had done that and I think it depends not he kid. If he is easily stimulated, it probably isn't a good idea if you want him to sleep.
After a few weeks of this consistent routine many kids resume taking naps every day. If he is ready to give up naps, he won't fall asleep after an hour of quiet time anymore and though he might go through some crabbiness in the evening while he's adjusting, he'll be able to make it until a reasonable bedtime.
Good luck~