Leaving out the potentiometer will not damage your processor or the LCD.
The potentiometer adjusts the contract of the LCD. If you leave the contrast adjustment pin unconnected, you might have a wrong contrast. Without the adjustment, the LCD could be all blank (white letters on white background), or it could be all black (black letters on black background). You may end up with the LCD, which is not readable.
P.S. A while ago, I went through the exercise of connecting the contrast pin incorrectly. Here's the thread. At first, I thought that this is a firmware problem. The realization that the contrast pin was the problem is in the last couple of posts in that thread.