Well I do Audio visual stuff alot. IE using lots of burning programs.
The only way you couldn't finalize a disk is if you were using CD-RW, CD+RW or the rare CD-RAM which can be used hundreds of times. Anyways if you didn't finalize these burns and they are still open well a couple of things could be happening because they should still play.
For example alot of expensive DVD players have protection on them to try and prevent pirated movies from playing on them. They won't play ANY written movies. For example I have about 6 DVD players in my house. My Sony will play nothing that doesn't have a AAC signature on it. So no burned DVD's. Of course that is my living room one so if I want to play a "burned" dvd I have to bring out my cheap one I got from target for 30 dollars that plays everything because it has no protection on it my Sony blu ray player will have nothing to do with them.
Keep in mind alot of the big name players also have ties to the MPAA and they don't want people downloading movies, or coping them illegally when they rent them... well because they put them out! So they have several ways of knowing if something came from the manufacturer or if you just copied it. Kinda like how Disney had a special coding on VHS tapes that prevented them from being copied ever. So this has exsisted for a long time.
Anyways what I'm trying to say is well you might want to try and buy like a cheapy DVD player and see if it will play those movies. Look for one that DOESN"T have wifi on it as well because those access the internet half the time and look for that signature as well. If you can find one that plays DIVX you are pry golden!
I'm not saying by the way what you have is illegal. For all I know these are home movies and such. But it doesn't really matter because like I said alot of these players if they don't see the AAC signature or something of that sort they won't play it. I think 4 of my 6 dvd players won't play anything "burned." Just the way it is.
The other thing I can think of is that the format that this old player converted to and played in isn't compatible with this player. Like if it converted your movies and burned them in AVI instead of NTSC and this new player isn't combatible with AVI it isn't going to play these movies. You'll have to find another player that is AVI or whatever format it converted in was.
Problem is I can think of several reasons why they can't play and most of them can be rectified one way or another but I'd have to see the DVD's and see what they are. BUT keep in mind if you choose to bring these into a professional and they are "pirated" movies or tv shows they aren't going to help you. Just a fair warning. Anything with a copy write from TV shows to movies that are burned illegally they aren't going to touch more than likely with a ten foot pole.
If they are CD-RW or CD+RW and you have no players that play them and you don't wish to buy one and they aren't finalized then you can go ahead and re-write them with something else. BUT they might still be worthless to you but you could put pictures on them and other such things you wish to keep. Just make sure you finalize them if you put your precious pictures on them. Most programs will give you the option if they are re-writable discs.
Good luck!