All creatures are subject to the adverse effects of having a "0" in any of their ability scores, but it should be noted that having a "0" is very explicitly not the same as having "-" or "N/A" in that score (that is, a creature with Intelligence 0 suffers averse effects from having a 0 in their Intelligence, but a creature with Intelligence - cannot suffer such effects. See "Ability Scores" in the System Reference Document). Some objects have ability scores, and strictly speaking the matter of if they're affected by it is up in the air; however, Undead are not objects in the game rules, but rather explicitly "creatures".
There's not a lot of effects actually capable of dealing ability damage to an undead's mental scores. They're immune to a lot of spells that would do it because those spells are [Mind-Affecting], and they're likewise immune to poisons and frankly almost all other means of inflicting damage of this kind. The only methods to harm undead in this fashion that I can recall are Positoxins (Libris Mortis), Ravages (Book of Exalted Deeds) and Afflictions (also Book of Exalted Deeds), and I'm not certain that both Ravages and Afflictions work - I think one of them (Afflictions if I recall correctly) still straight-up does not work. As Ernir pointed out in comments, Ravages and/or Afflictions note an specialized effect on undead, but fail to make a specific exception for the normal Undead immunity to effects requiring a Fortitude save; it's safe to assume that they were intended to work, but by RAW the lack of the crucial exemption clause indicates that they don't. As always, bring it up with your DM to get a ruling if you intend to use those avenues of attack.