From Diablo Wiki
There are no immunities in Diablo III. Monsters and players have resistances to types of damage, but nothing in Diablo 3 is immune, meaning they take no damage from any particular type of attack.
This is a big change from the combat mechanics seen in Diablo 2, where dealing with monster resistances and immunities to all sorts of elemental attacks, and even physical damage, was a major challenge for characters on the highest difficulty levels. This change in Diablo 3 went along with the general de-emphasis/removal of elemental damage types from weapons.
Though weapons and skills (especially Skill Runes) list their damage in types such as, poison, arcane, fire, etc, these are purely cosmetic (except for cold, which may chill/slow targets) changing only the visuals of the attack and the death animations of the monsters.
The removal of elemental damage types and monster resistances/immunities remains controversial, and is often cited by dissatisfied players when they complain about Diablo 3's "dumbed down" design decisions.
 Immunities During Development
Jay Wilson: Resistance was changed from a flat percentage to a basically a rating, that works how defense did (in Diablo II). It scales with the level of the monster. So it can grow to higher values over time. With this change we do not have to include resistance penalties for Nightmare and Hell, as was done in Diablo II. In a lot of ways our resistances are the same. Resistance is still a big part of the game. It's a focus of your defensive building of your char in higher levels.
As for immunities, we're not doing double/triple immunities (in monsters). We don't want to completely hurt a character's ability to hurt something. We also diversify damage types more across the classes (than was done in Diablo II). But we still can't count on a class having a diversity of damage types we can use.
Will characters need to stack resistances for particular fights?
Jay Wilson: Sometimes. It's kind of an end game question, so haven't done it yet. We'll look more at it as the time comes closer.
The thinking had changed by 2012, and Jay confirmed that immunities were out during a press appearance in May 2012.
Jay Wilson: We do not use immunities as a monster affix. We focused more on powers that change up the monster’s threat and challenge different classes in different ways.
 Character Skills
A few character skills showed an ability to break immunities during development, but no skills or items do this in the final game since there are no immunities to break. One quote that was noted from a skill description at Blizzcon 2009 can be seen below:
- Description: Enter a heightened sense of fury, raising several stats (Critical Hit, Attack Speed, Dodge Chance, Movement Speed) for X seconds. Monsters with damage immunities will take X% less damage from the Barbarian's attacks rather than being immune.