How did you install it? CD? DVD? Flash drive?
To your internal hard drive? One hard drive or more?
When your installation finished, you should have seen a message indicating so and asking you to reboot, did you see that? After rebooting, did you then see a message telling you to remove the disk and hit the Enter key? Have you tried the eject button? What happens? Have you tried shutting down the computer and rebooting? What happens? What exactly have you tried?
I don't see how installing Mint or any other OS will cause your drive not to open but I'm not a hardware expert. I've had my DVD drive fail to open any number of times and often for no discernible reason.