Okay, time to be a bit of a pedant. Not only can you run DMX on XLR, but you do every single time you use it in most any application.
XLR is NOT a type of cable, it's a type of connector. DMX by spec is on 5-pin XLR connectors, while it sometimes is implemented with 3-pin. Audio signals also use 3-pin XLR, on what's often called microphone or audio cable, a particular type of shielded twisted-pair (STP) cable.
What you mean to ask, I presume, is "can you run DMX down audio cable," and as others have posted, the answer is that you can, and it may work, depending on circumstances and even a bit of luck, but you shouldn't. There are very specific characteristics of digital-grade cable that typical audio cable doesn't meet, and as noted, it can degrade the signal.
It's one of those where if it works, great, but if it doesn't (and it might at first, but then stop later on), there's nothing you can do to fix it.
You can, for example, run analog audio down digital grade cable, but the reverse is not a good idea. It's sort of a square vs. rectangle thing, y'know? This is why a lot of installs today are spec'ing digital cabling for all audio lines, so that if later on somebody needs to run AES/EBU or another digital audio signal down it, the proper cable's already in place. A bit more expensive, but cheaper than having to re-install cable later!
Here ya go:
I didn't see anything that even mentioned cat5 cable on there, let alone pinouts...