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An antimagic field spell makes magical weapons into normal weapons for purposes of attack. Magical spells also lose their effects.

A lich is listed as having damage immunities to: Poison; Bludgeoning, Piercing, and Slashing From Nonmagical Attacks. Liches also have condition immunities to: Charmed, Exhaustion, Frightened, Paralyzed, and Poisoned.

The only effects that penetrate an antimagic field are artifacts and effects from deities. Given the immunities of a lich, this seems to leave only fire and acid as a way to damage liches apart from artifacts and deities.

However, Jeremy Crawford previously wrote, "Antimagic field has no effect on a creature's immunities, unless those immunities are sustained by magic."

Are a lich's immunities "sustained by magic" as described by Crawford and go away in an antimagic field? More generally, apart from artifacts, deities, and fire, what can damage a lich in an antimagic field?

The lich retains its immunities but can still be damaged

The lich's immunities are not sustained through a magical effect or spell which are what anti-magic field suppresses. They are inherent to the lich and are not mentioned as being "magical".

That said, there are still ways to damage a lich, for example Bludgeoning/Piercing/Slashing damage that is not from an attack. For example, the catapult spell may work (discussed here) or fall damage. Also as user @guildsbounty points out, objects falling onto the lich will also work, perhaps you could make the ceiling collapse. Another method is non-magical elemental damage which is available through various features and items such as alchemists' fire. This answer to the question "Does every damage type have a way to deal nonmagical damage?" shows ways for PC's to have non-magical elemental damage.

Class abilities and certain items

Alchemist's fire, vials of acid and holy water are all non-magical items which can cause damage the lich does not have immunity to.

Certain classes gain abilities that can allow them to damage a lich even inside of an antimagic field. Some examples:

Monk's at 6th level gain the Ki-Empowered Strikes ability which makes their unarmed strikes count as magical for the purpose of bypassing damage resistance and immunity. This is not actually a magical ability so the AMF won't stop it.

Clerics who have the life domain gain the Divine Strike ability at level 8 which makes their weapon attacks do extra radiant damage.

Paladins gain Improved Divine Smite at level 11 which gives them extra radiant damage on their melee attacks.

Warlocks with the Fiend pact gain Hurl Through Hell at level 14 which can send the lich through the lower planes and make them take psychic damage on their return.

There are a few skills/class features that should still work

For instance, a 6th level monk gets "Ki-Empowered Strikes" which:

Starting at 6th level, your unarmed strikes count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

Similarly, the Warlock's Pact of the Blade:

You can use your action to create a pact weapon in your empty hand. You can choose the form that this melee weapon takes each time you create it (see chapter 5 for weapon options). You are proficient with it while you wield it. This weapon counts as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

Even the lowly Beast Master Ranger, at 7th level gains Exceptional Training:

Beginning at 7th level, on any of your turns when your beast companion doesn't attack, you can use a bonus action to command the beast to take the Dash, Disengage, or Help action on its turn. In addition, the beast's attacks now count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

All of these examples are not magical so they won't be suppressed, but will still overcome resistance and immunity.

I know this list is not exhaustive, but should be a good starting point.