From Diablo Wiki
Special player death animations were a long-promised feature in Diablo III, but do not appear in the final game.
Fatalities were first teased in 2008 in the first gameplay movie from WWI 2008, when Siegebreaker messily bit off the Barbarian's head, as well as picking up, holding, and then spiking a Witch Doctor. More Blue comments were made on them over the years, building fan anticipation for this feature.
Unfortunately, in January 2012 the developers admitted that they had never gotten implemented into the game and would not be there upon release.. This feature remain on the developers' wish list, and might be added in an expansion pack.
FatalitiesSiegebreaker inflicted upon one of the Barbarians in the WWI Gameplay movie.
Will there be more boss-specific fatalities like the Barbarian head bite by Siegebreaker?
Bashiok: That's what we would like to do. We'd like it that when a player dies to a boss we may have a special/random event or death animation specific to that boss. So if you're going to die or have died to a boss, there's a random chance that you'll see something other than a normal death. That's the dream any way.
Jay Wilson: Well what we want to do with that, especially for particular bosses is have when essentially when the player is very low on health and they're about to die the boss essentially checks like when it attacks you, did I just do enough damage to kill you? And if so then instead of just doing his normal attack he'd actually play some kind of special I pick you up and eat you or I throw you up in the air and knock you around like a baseball or something like that. It's a system that really our announcement video we tested it for the first time, mainly just to see – can we actually do this kind of animation interaction between the characters but the actual system itself is still not in there but we do plan to do that and that's primarily where. We might do some other things, we've talked about physics based deaths we've talked about having the characters if they like die to a cold monster he might completely freeze solid into a statue and then shatter or things like that but we haven't decided at this point if that's exactly what we're going to do.
Unfortunately, this feature did not make it into the actual game, as revealed in January 2012. 
All things considered there are just higher priority tasks sometimes. We’d love, for example, to have the monk display his weapons during all of his skill animations. But it’s an enormous animation and effects investment to get it done, and we have to weigh all of these things against other features and polish. A lot of the skills we could improve have multiple animation sets, multiplied twice for each gender, and then multiplied by each weapon type he can equip. That can be 40 or more unique animations, and then there’s a full FX pass, approvals, QA testing, etc. etc. and that’d just be for 1 skill.We have our wish lists, but it’s a balancing act of resources. Ideally our choices when balancing go toward setting realistic limitations, and producing the best game possible.
- Bashiok forum post - August 2009