When I plug unit in, I can hear the motor run for about 30 seconds. The CD drawer opens and closes fine but after it opens and closes, the motor keeps running. When a CD is put in and close button is pressed, it closes but won't play a CD and spinner won't turn.

I cannot figure it out. Is it possible the belt on the door motor needs to be replaced? Is it just not sensing the drawer is all the way open or closed somehow?

It might be a belt, it might be a microswitch or something. No way of telling without opening it up and having a look 

Could be a lot of things. Need to disassemble and look what the things are doing in detail.
Could be a dirty switch, loose belt, dusted up optics, weak laser,... Troubleshooting is needed in which phase it get stuck.
Normally it should:
-Close the door
-Spin up the CD
-move the optics carrier back and forth few times to find the initializing track
-At the same time light up the laser (VERY dark and weak tiny red dot "hovering" above the lens when looking directly into it, nearly invisible to an eye, so not much sense to explicitely look for that by eyes)
-At the same time moving the lens up and down (an attempt to get into focus so some signal from something on the CD)
-If the CD is in and everything works, after 2..3s it starts playing or displays the number of tracks (depend how it is programmed to act). When no CD is there or is unreadable (damaged, or there is something wrong with the optics like dusted or so), after about 10..30s of shuffling it gives up and eventually throws an error message (again depends how it is programmed).

Thanks all, I think I figured it out.

So I took the cover off, and I found if I put my finger on the motor spindle, after the drawer opens or closes, that the motor would stop running.

After I touched that after the drawer was closed, I found that the CD would start spinning but it still wouldn't play anything and it just said "ND."
So I took the CD tray out, unplugged all the connections, and moved the worm gear that the laser moves on back and forth. Then put it back together
plugged all the connections back in, and I was able to get it to play a CD. Not sure if it was a lose connection or the worm gear was stuck. But it does play now.
One of the connectors did look a bit loose so maybe that was the problem.

Of course once I put it back together, it won't play without me stopping the drawer motor spindle first with finger.

I think the problem there is it just needs a new belt, as it is a bit loose. I think there's just not enough tension
for the motor to sense that the drawer is fully open or closed, so it keeps running and the belt slips.