How Can I Mod a Xbox 360 to Be Able to Burn Games?

You cant. End of story. The xbox lacks the required hardware to do this. You'd have just as much luck trying to get a toaster to build you a spice rack.

Suggestion:

a game copy software plus a blank dvd are capable to make it. i used a software to burn my xbox 360 games, the sound and picture quality of the resulting copies are just like the originals’. here is it, hope it may help you:)

http://www.copygames.org/guides/rip-xbox-360-games.html