I shoot both .243 & '06. I hunt mule deer with my '06. If I could hunt from a stand, and went only for neck shots, I wouldn't mind using my .243.
It's not that the .243 won't work. It is that the '06 has more punch in the event of a less than perfect hit.
Lots of mule deer have been killed with everything from .30-30 on up. .270 is fine. .25-'06 oughta work. The thing is, for a really worthwhile mulie, you want to make sure you get him cleanly and "for sure". Nothing makes you feel worse than hitting something and having it get away, to die a day or three later...
The last mule deer I got was offhand at 30 yards. I shot him in the neck with the '06, using my 150-gr Sierra load. The bullet did not exit. He was paralyzed, not going anywhere, but definitely not dead! He only field-dressed at 150 pounds...
So, unless you can regularly hit the end of a beer can at 100 yards, and have satisfied yourself on an occasional running rabbit, I'd recommend against the .243 as your primary selection.