A restrained lich in an antimagic field-warded cell may find suicide impossible...

Because of its undead type, I'm not altogether sure a lich, if somehow captured, restrained, and confined to a prison cell warded with an antimagic field, can kill itself without outside aid.

A lich doesn't breathe, so it can't hold its breath until it dies. It can't hunger strike to any real effect as it doesn't need to eat to live. It may dehydrate, but starvation and thirst in the SRD deal nonlethal damage to which a lich is immune. It can't be poisoned or catch diseases from prison vermin, other prisoners, garbage, or whatever. Because they require Fortitude saving throws and don't affect objects, ravages and afflictions (BE 34-6) don't affect undead creatures: even if the affliction haunting conscience sweeps through the Dreadhold, a lich won't be violently angry and dwell on its evil deeds so that it suffers Wisdom damage.

A lich's damage reduction is explicitly supernatural, so its damage reduction is suppressed in the antimagic field-warded cell. But a lich possesses no special ability to spontaneously kill itself (be it through self-neck-snapping, swallowing its tongue-remains, or whatever—if any of those even work for a lich); if it desires its own destruction, it must do so through brute force. This means a determined, unbound lich can claw itself or unarmed strike itself to death without issue, making restraints a necessity if one wants the lich's imprisonment to persist.

Even a bound lich (or, for that matter, any creature) with ranks in the skill Lucid Dreaming (Manual of the Planes 203) can hurl itself into the Dreamheart, possibly leading to its death, but doing so requires using a variant planar cosmology that some consider unbalanced. (This is because a lucid dreamer instead could've used the skill Lucid Dreaming to gain enough intelligence—or to kill or threaten to kill enough people—to escape its confinement rather than using the skill merely to kill itself) (This also assumes a creature that need not sleep—like a lich—can nonetheless choose to sleep; ask the DM.)

Wardens should be aware of the spell invoke magic

An extremely careful lich may know or prepare the following spell:

• The 9th-level Sor/Wiz spell invoke magic [evoc] (Lords of Madness 212) for 1 round allows the caster to cast 1 4th-level or lower spell in an area where magic doesn't normally function, such as within an antimagic field. The spell invoke magic has a casting time of 1 swift action.

A paranoid lich wizard may even take as its level 18 feat Spell Mastery just to be able to prepare without its books this spell (and also, of course, like, dimension door or another get-me-out-of-this-antimagic-field-warded-cell spell).

An alternative to incarceration

The 7th-level Drd spell rain of roses [conj] (Book of Exalted Deeds 105), among other effects, deals evil creatures within its cylinder 1d4 points of Wisdom damage for 1 round/level, yet the spell isn't mind-affecting, so it's capable of forcing a lich to become "withdrawn into a deep sleep filled with nightmares, helpless." A spell clock (Clockwork Wonders Web column "Part 12 of 14: The Spell Clock") (130,000 gp; 0 lbs.) or an automatically resetting magical trap (like the earthquake trap) could at regular intervals continue such an effect. However, getting a lich into a position where this could occur is likely an adventure unto itself; interrogating such a helpless lich is, so far as I can tell, impossible; and taunts aimed at such a helpless lich will fall on deaf ear-remains.

...And such an incarcerated lich will soon find itself interrogated or taunted

Theoretically, any creature emotionally, physically, or magically attached to a thing that can be destroyed (like, for example, a lich to its phylactery) can be made to spill secrets by threatening that thing with destruction. Determining then acquiring that thing becomes the hard part, often the crux of one or more adventures.

Torture (Book of Vile Darkness 37-9) appears to work normally on a lich unless the DM rules that undead creatures are immune to Intimidate skill checks because they're immune to mind-affecting effects. As this could also also mean no one can ever employ against a lich the skills Bluff or Diplomacy, this is a difficult ruling.

Besides that, there's magic—except such magic can't be mind-affecting; require a Fortitude saving throw unless the effect also affects objects (rendering useless, for example, flesh to stone); be a poison or a disease; drain an ability score; damage a physical ability score; be a paralysis or death effect; involve dealing nonlethal damage; inflict negative levels; deal cold or electricity damage; or be a polymorph effect. This window is so small, I recommend widening it with the following:

• The 3rd-level Clr spell spark of life [necro] (Spell Compendium 196) causes a touched undead creature to make a Will save and check SR. Success means no effect. Failure means for 1 round/level the undead creature is no longer immune to critical hits, nonlethal damage, ability drain, physical ability damage except Constitution damage, energy drain, fatigue and exhaustion, poison, sleep, paralysis, stunning, disease, death effects, and effects that require a Fortitude saving throw. It must also breathe, eat, and sleep. It gains a bonus on Fortitude saving throws equal to its Charisma bonus against all effects except those effects also affecting objects. Negative energy effects that would heal the creature (like inflict spells) and positive energy effects that would damage the creature (like cure spells) both heal the creature.

So while an undead creature affected by the spell spark of life remains invulnerable to mind-affecting effects, if a caster can't get a lich in an antimagic field-warded cell to squeal while the lich is vulnerable to all that other stuff, that caster should give up on his career as an interrogator and, like, go craft spell component pouches or rear future familiars or something.

Note, however, that the spell spark of life also makes it far easier for the lich to kill itself or to be killed accidentally, and to cast the spell spark of life on the lich (unless also using invoke magic or a like effect) requires removing the lich from the antimagic field-warded cell. Caution is advised.

"Let's just keep its head"

According to the description of the magic weapon special ability vorpal (DMG 226) (+5 bonus; 0 lbs.) "undead creatures other than vampires... are not affected by the loss of their heads." Maybe the lich's head can be safely kept in a jar? Maybe that also means the lich's (maybe blind, maybe deaf) body hunts for its head, headless horseman style? This is so wild only the DM can tell you what happens.