Yes and no.
I will assume the data burned to the disc okay, and it has been verified to not contain any bad sectors.
The problem with many lower-grade discs is that they are harder to read by devices -- indeed, this is one reason they rank lower than discs such as Verbatim, Sony or Taiyo Yuden.
If your device has trouble reading the disc, most often it will simply give a big "fail" or "bad disc" message on screen, sometimes even ejecting the disc. This is especially true if the disc is hard to read at the start.
If your device -- in this case, an XBOX 360 -- wants to retry and retry, it can wear out the laser assembly in time. This is most commonly the response of a device when the error is found in the middle or towards the end of the discs. If you do this a few times with a movie, it won't necessarily cause any damage. It's still not good, however. If the player/device is constantly subjected to media that it must try harder to read, then it will prematurely kill the machine.
If this is a video game or some other constantly-used disc, I would not use anything but a top-quality blank. Review the guide for a list of the best media to use, and where to buy them:
http://www.digitalfaq.com/reviews/dvd-media.htm
So, yes, it is possible to damage your XBOX 360 with cheap discs, in the long-run. This is why you find so many drives dead in PS2's and XBOX's from years past -- especially from the stubborn asses that insisted "I can use any disc I want, and it works fine". Those folks would play burned games and movies in the game systems, and it eventually took a toll on the DVD drive inside, killing it. That was also true of the first Playstation, and cheap CD-R.
Also be aware that Teac discs have, in the past, used fake media IDs, such as fake TYG02 Taiyo Yuden codes. Those 8x Teac were NOT Taiyo Yuden media. Teac is a bottom-barrel brand that uses all kinds of mediocre or outright crap discs.
