The paladin may close his eyes. However, he still needs to be facing a target:
From the SRD:
Detect Evil: At will, the paladin can detect evil as a spell-like ability. This ability duplicates the effects of the spell detect evil.
Range: Quarter circle emanating from the character to the extreme of the range
So it acts more like a flashlight than a radar, per se. Of course, if he wants to match person to aura, he'll need to open his eyes. In a crowded room, without maintaining concentration, he'll loose them. And while it can go through walls, not as far as you might think, and not stronger walls:
from the SRD:The spell can penetrate barriers, but 1 foot of stone, 1 inch of common metal, a thin sheet of lead, or 3 feet of wood or dirt blocks it.
For example, in "Heart of Nightfang Spire", a Paladin's Detect Evil ability would make finding the baddies pathetically easy...if not for the thick, stone walls blocking his detection ability.
Furthermore, unless the Paladin is
not known to be a paladin, people will most likely assume the worst when dealing with him. I would expect a circumstance penalty to reactions if he did this in public and was noticed. Even if all he does is simply concentrate for a moment...if someone talks to him and he suddenly seems distracted, it could be problematic. Of course, he may not even be invited to such functions if he has a reputation of doing such things.
Which isn't to say he shouldn't be allowed to detect evil in such a situation, but it is somewhat rude. It also never hurts that Good and Neutral characters can be antagonists, without being villains. My party just dealt with a LG antagonist recently, for example.