As Rudegar said, it's really up to you and how sensitive the data on the drive might be. If you had FileVault set up, then there is no need to remove or otherwise worry about the drive. If not and there's sensitive data on it, such as plain-text copies of your account passwords, then erasing the drive becomes more important. You can see if perhaps you can get that old Mac into Target Disk Mode:
Transfer files between two Mac computers using target disk mode - Apple Support
If so, then you may be able to mount that drive on your new Mac and erase the volume. If not and the information is truly sensitive, you can disassemble the Mac and remove the drive and then either find an adapter to connect it so it can be erased or just smash the drive with a hammer.
But before you do all that, you might just ask the Apple Store what their process is for recycling a Mac. I'm sure they've dealt with the issue of remaining data on a Mac when the Mac won't boot, so they may be able to ease your mind.
Regards.