Why would an internal DVD reader eject a burned DVD?
I noticed on some MacBooks and MacBook Pro(s), that the internal "media/super-drives" are very picky, especially when it comes to DVDs and other problems, which in this case are not much relevant. (Individual user experience maybe?)


You can always try to use a different DVD brand/producer. That might do the trick, at least in my experience. 
It sometimes is a good idea to use "low burn speed rates", depending on the contents you are going to burn and also to use buffer settings within the application itself, to prevent faulty sectors/errors.
Though it shouldn't be a problem at all, it is a good idea to use common applications, like "Disk Utility.app" or "Toast.app", which are common on the other users system too.