I've summarized below what I think are the major issues raised by the question. I hope that's okay.
Can an attacker close its eyes to avoid the effects of the mirror image spell?
Yes, but this doesn't automatically reveal to the attacker which square it's target really occupies and the typical attacker's attacks suffer a 50% miss chance because the attacker's usually now blind.
Does the ability scent let the creature differentiate between figures on the battlefield that are figments created by the mirror image spell and the caster of that mirror image spell?
A typical creature with scent knows that a creature has an odor whenever the creature with scent becomes adjacent to a creature (like through movement). A figment created by the mirror image spell has no odor.
A creature that possesses extraordinary ability scent (Monster Manual 314 et al.) can take a move action to learn which direction other creatures are in if they're within the range of the first creature's scent. Knowing direction, though, doesn't also tell that creature what squares are occupied by those other creatures. The DM just tells the creature that other creatures are, like, over there. The creature with scent must become adjacent to another creature to pinpoint another creature.
In D&D 3.5, the term pinpoint is an exaggeration. A typical attacker that can't see the target (like, for example, the attacker has closed its eyes or is blinded) and that has only pinpointed a target (like through a successful Listen skill check or by using the scent ability) suffers a 50% miss chance against that target. (See here for how that works; blinded uses darkness mechanics.) Also, some controversy surrounds a creature closing and opening its eyes, like how often and how quickly, but this DM hasn't had any issues with characters taking free actions multiple times per turn to do either. This DM doesn't allow a creature close or open its eyes off-turn, though. (See also gaze attacks (MM 309–10).)
At the gaming table, it may be easiest to represent figments created by the 2nd-level Sor/Wiz spell mirror image [illus] (Player's Handbook 254) with figures identical to the spell's subject. See, the mirror image spell "creates 1d4 images plus one image per three caster levels (maximum eight images total) [that] separate from you [i.e. the caster] and remain in a cluster, each within 5 feet of at least one other figment or you," and the "figments mimic your actions."
So the figments are not, for example, overlapping with the caster (at least initially) and doing their own special things to distract his foes. Instead, each figment is a duplicate that occupies its own separate space on the battlefield and copies the caster's actions. (A caster can later opt to share a mirror image spell's figment's space, but this doesn't affect this answer.)
Thus when a creature that possesses the extraordinary ability scent becomes adjacent to one of those figments, the creature will automatically learn that the figment is odorless. (For any figment to possess an odor, the description of the figment-creating effect must say that the figment possesses an olfactory component like the image that can be created using the 5th-level Sor/Wiz spell persistent image [illus] (PH 260).)
This DM would count this as interacting with the figment—getting adjacent to a thing is one way a creature with scent interacts with things with its scent, after all—, but even on a failed saving throw to disbelieve that figment, the creature is still allowed to act on its knowledge that the thing nearby is odorless, possibly going so far as to ignore that odorless thing. This can be risky, by the way, in a game that includes the 1st-level bard, druid, and Sor/Wiz spell remove scent [trans] (Spell Compendium 173) and that contains creatures like allips and shadows.