My MacBook Pro is popping out DVDs

Lately my mac is rejecting some DVDs, including a double-sided Mortal Kombat and a couple of DVDs I made ON this very Mac before I replaced the hard drive.

Could the drive be failing?

Hello, Welcome to M-F!

Usually this kind of a problem indicates some kind of a problem with the DVD drive, so don't be to surprised if you have to have it replaced.
While it could very well be that the DVD drive is failing, it could also be that the optical sensor is dirty in some way.
My brothers MBP had a similar problem once, so we got a 5$ DVD drive cleaning disk from Staples and let it run in the drive for 5 minutes and it seemed to work after that.
Amazon.com: Memorex CD/DVD-Player Laser-Lens Cleaner: Electronics

If you have AppleCare take it to Apple, won't cost you anything to have it checked out, and if it needs replacing and you don't want to do it yourself they will do it for you.

Welcome, and enjoy your stay. I would agree with McYukon on the dirty optical sensor. My dad's Dell laptop for work was doing that with videos he needed for presentations. Long story short I used a drive cleaning disk similar to the one in the above post and afterwards it worked just fine. It wouldn't hurt to do that first, and if it doesn't work (if you're still under Applecare) you can still call Apple and get their take on it.