Can a XBOX 360 arcade play games without a DVD Drive

Yes and No.

You would not be able to play games from a disk but you should be able to play games installed to your hard-drive, these hard-drive games would be limited to arcade and demo downloads ONLY. Any games you have installed from your disks (assuming you are connected to XBL and have the new update) would not be playable as the console needs the disk inserted to boot the game up.

This is assuming the drive has broken and not actually been completely removed, In either case using your 360 is a bad idea and can only cause more damage, possible fire and electrecution risks would probably also be an issue.