My 2 year 11 month old has not napped really since last summer, and even then it was sketchy. But like others here, it would go in spurts. A few months ago it got a little more stressful going through the whole process, since I would do a mini bedtime routine with her (milk, stories, pray, songs). She got a stomach bug right after Christmas when we were out of town and after that she wouldn't drink milk anymore...and that was sort of the trigger for her nap/bedtime. She gets bedtime, even if she doesn't want to go, but would definitely try to distract us from naptime. So, after a few weeks of struggling over the whole routine, I finally just decided to call it quiet time. She seemed to feel better knowing that she wasn't going 'night night' and would just read books quietly in her bed. Since January, we've gone through a number of phases with her getting out of bed and playing to taking random 2 hour naps. Since I have a 5 month old, I need her to have some time to rest--even if she doesn't sleep, she is usually in a better mood after she has the time alone. All of that to say, if you're not bent on him sleeping, perhaps you can persuade him to just have a little down time without the expectation of sleeping. He may be more receptive. My husband likes to have my daughter change into her pajamas, like we just to do for her nap time, but I don't always do it. But I do try to keep her as quiet as possible and in her bed. We have a video monitor so we can watch where she is at--this whole thing is part of the reason why we put up a video monitor in the first place! We couldn't tell if she was out of her bed or not and it was getting frustrating. I often put books on her bed so that she can look through them, though I don't leave the lights on. In the end, even if she is singing or playing, I need the time either to myself or to take care of the baby, so I don't stress too much over the details. But...we do put her to bed on the earlier side. Anywhere from 7 to 8:30, depending on when we have dinner, etc. Our routine is always the same, but if my husband works late or if the baby requires more of me, then it seems like everything just get pushed back. My daughter does sleep 11.5-12.5 hours at night as well, so I know she is at least getting the minimum amount she needs.
Unfortunately, lots of kids don't keep the nap as long as we'd like, but getting him to do a rest time would be good, even for when he goes to school eventually. Hope it gets better for you!