A man who is on the run or otherwise trying not to be recognized will often grow a beard as part of his attempt at concealment.
The inversion -- where a heavily bearded character shaves for similar reasons -- is also relatively common.
- Duncan grows one near the end of the first season of Veronica Mars. Unless it's a Beard of Sorrow; at the time that he grows it, he's both depressed and running away from home.
- In the film version of The Fugitive, one of the first things Dr. Kimble does after escaping from prison is shave off his beard.
- In the Lord Peter Wimsey book The Nine Tailors, Nobby Cranton grows a beard before going to look for a diamond necklace he'd stolen some decades before.