Monsters and pets can be immune to any of the six damage types when Resist > 99%. Any inherent immunities are displayed as labels below their life bars, although those of pets can only be displayed to the summoning player if that player can cast Holy Bolt on them or they can be unsummoned, by selecting Unsummon on the right mouse button and placing the cursor over the pet (with the exception of Decoy, Shadow Warrior and Shadow Master, which will only display the life bar over their head rather than at the top of the screen with any labels).
Pets can also become immune to more damage types if inherent resistance and bonuses from skills and their own items (in the case of Valkyrie, Shadow Warrior, Shadow Master or Iron Golem) result in Resist > 99%. However, these immunities will not be displayed unless they include inherent resistance, or an individual skill or all items apply Resist > 99%.  
This level 15 Valkyrie (All Resistances
+32 with level 1 Decoy
) is Immune to Fire, Lightning, Cold and
Poison due to resistance bonuses from her items. Although none of this is displayed, she takes no damage from a barrage of missiles
and poison damage
, and she is neither chilled nor poisoned.
Immunities dependent on Summon Resist are never displayed, even when they include inherent resistance. 
Monsters could be inherently immune to all six damage types, although in practice only one is inherently immune to more than two (Pit Lords of the Pandemonium Event are always immune to all but physical damage). Inherent immunities always apply and they are always displayed.
A small number of monsters spawn with items whose randomly selected affixes may apply resistance bonuses which can result in them becoming immune when added to inherent resistance. Immunities dependent on items are not displayed.
is not poisoned by Rabies
in Normal as a combination of his own Poison Resist
70% and PR ≥ 30% from item affixes makes him immune.
Monsters may also become immune due to resistance bonuses applied by Unique bonuses (which only apply to the monster itself, and not to any minions).  However, these resistance bonuses only apply if the monster has less than two immunities beforehand as a result of inherent resistance, bonuses from items and earlier Unique bonuses   (they are applied in the order in which they are displayed below the life bar, from left to right).
1 Poison Enchanted label is not displayed
2 Spectral Hit alone can never result in immunities, but its bonuses can increase those resulting from later bonuses
Although Unique resistance bonuses are not applied if there are two or more immunities beforehand, if there are less than two without item bonuses then those displayed are any inherent immunity and any applied by the Unique bonus(es).
is not damaged by Fire Ball
in Hell as his FR 50% and FR ≥ 50% from items makes him Immune to Fire. He is also inherently Immune to Lightning
chilled by Glacial Spike
in Hell despite displaying Immune to Cold, as he is already Immune to Lightning, and Immune to Fire due to his inherent resistance and items.
Bosses of the Pandemonium Event have Unique bonuses and immunities which are not displayed. Ghostly champions are set to Damage Resist 80%, even when Immune to Physical; however, the Immune to Physical label remains.  
Although the Barbarian War Cry
's physical damage obviously doesn't damage the Immune to Physical Champion Blood Lord
, it does
damage the 'Immune to Physical' Ghostly Blood Lord
since it has actually been set to DR 80%.
Download Monster Resistances.xls for a filtered list of all monsters and their inherent resistances and immunities in Hell; if you don't have Microsoft Office or Excel, you can download Excel Viewer or OpenOffice for free to view it.
Skills applying Resist -% can result in Resist < 100%, removing an immunity and allowing damage of that type to be applied. However, whenever an immunity is inherent or a Unique resistance bonus contributes, the sum of Resist -% applied is rounded down to 1/5 even if item affixes result in immunity, or they are required for a subsequent Unique bonus to do so. For example, Resist -41% from level 3 Lower Resist and Resist -85% from level 12 Conviction applies Resist -126% in total, which is reduced to Resist -25% [126/5] when applied to an immunity (even when either used in isolation would be sufficient to remove it).
When immunity is removed, any label beneath the life bar normally disappears and any relevant -% Enemy Resistance equipped now applies at full effectiveness. However, a label may be removed even if item affixes mean the immunity is not: for example, Talic's inherent fire immunity in Hell may appear to be removed by level 10 Lower Resist (110 - [55/5] = 99), but if he spawns with Crimson Myrmidon Greaves (Fire Resist +5%) or better then this does not actually happen (110 - [55/5] +5 = 104).
Skills applying Resist -% are still fully effective when applied to immunities wholly dependent on skill or item bonuses.  For example, an Iron Golem is normally Immune to Poison with Poison Resist 100%, and Lightning Resist 50%: making it from an item with LR +50% and All Resistances +50 from level 6 Summon Resist would increase these to PR 150% and LR 150%, making it Immune to Lightning as well; however, casting level 10 Lower Resist would remove its lightning immunity (150-55 = 95) while its poison immunity remained (150 - [55/5] = 139).