My son outgrew naps by 18 months and, to this day, needs far less than average sleep. I think trying to set aside quiet/alone time for him as others suggested is fine, but if he is also needy in the way of attention, as my son was, you may find that is far too traumatic to enforce.
BUT, you do need to set a bed time and teach him to stick to it. The more over-tired they get, the more hyper they get, and they really don't understand that this is what is happening.
Another complication is if your child has trouble falling asleep. It isn't so much that they don't want to go to sleep, but they don't know how to deal with the 45 minutes or so it takes them to fall asleep. With my son now 10, it is now evident that this, also, was a huge part of the issue when he was young. How many hyper toddlers want to stay in bed with nothing to do when they can't fall asleep?
A couple of things I tried, some of which I feel better about than others:
(1) I remember putting my son to bed when he was was 2 or 3 (we had a nice routine for that, with detailed imagery stories I would tell) and then pulling up a cozy blanket for myself and parking myself in front of the door so he couldn't escape. I used a dimmer to make the room appropriate for sleep. He would try to move me, wanting to play and leave, but I would keep repeating that it was bed time, we were going to sleep now, and he would eventually fall asleep in some strange position on the floor. While this was difficult emotionally, since I was with him, and the action was passive, it played out more as a battle of wills than anything that would cause him long term trauma.
(2) Sometime after we had crossed the "escape" barrier, I remember telling my son that I wouldn't make him fall asleep at all, it was up to him, BUT he had to stay in his bed and couldn't play with toys. A stack of books, however, was fine. I set up the baby monitor and then went in my own room to read. Eventually, he would put himself to sleep, and I would go in, remove any books on the bed, tuck him in, and turn out the lights. Over time, I was able to give him a limit on how long he could "read." While things kept changing in our family and there was a period I couldn't use this routine, what I like about it is that it taught my son a very useful skill, and at age 10 he still uses reading before bed as a way to decrease his insomnia.
I will also comment on the preschool suggestion, and note that having my son in a preschool with a required nap period was a disaster and one of my biggest regrets is that I tried for so long to make that work. It was literally traumatic for my son to be forced to be quiet and inactive for 2 hours when he could not sleep, and it created behavior issues. Some kids will follow the "group" think, and take a nap when they otherwise would not, but some kids like my son will not, and it became a huge issue for both him and the school. There was no flexibility there on it; he HAD to "nap."
In general:
Remember that any new routine will take a while to show if it's working or not, and remember that every child is completely unique, and you should never put your mommy instincts behind advice you have read from others. Sleep can be a strangely unique thing, and you want to help your child find the road to good habits that will work for him. Not easy at all with some children, but so it goes.