A legendary bow's name is "Etrayu". Now, exchange the letters "a" and "e": you get "Atreyu", one of the heroes of the The Neverending Story, whose weapon of choice was a bow. If you had any doubts about it, the item's description leaves no ambiguity:
The warrior surveyed the dead and dying arrayed about him. There would be moreómany more. He felt as if he were trapped in a tale that would never end.
There's a unique Ghastly Gravedigger with the name Digger O'Dell and subtitle "must be shoveling off." This is a reference to a 1940s radio sitcom called The Life Of Riley. Apparently nothing is too obscure for Blizzard!
The Monk has a skill called Exploding Palm. The key feature of this attack is that the victim explodes if the DOT kills them (later edited to apply to any damage while the debuff still applies). Four words only: You arealready dead.
There is a three-bladed legendary fist weapon called Logan's Claw that regenerates the user's health.
The female Demon Hunter bears a strong resemblance to Selene from the Underworld films, and a number of achievements and runes (such as "Death Dealer" and "Equilibrium") reference those and other movies with an emphasis on Badass Longcoats and Guns Akimbo. There's even a legendary demon hunter cloak called Beckon Sail.
Covetous Shen strongly resembles David Lo Pan in mannerism. Unsurprising, considering he is voiced by James Hong himself.
You initially find Covetous Shen trapped inside a barrel. If you wait long enough before rescuing him to hear all his dialogue, you get an achievement called "Big Trouble in Little Barrel" which just underlines the reference.
One set is called Chantodo's Legacy. One piece of this set, the orb, is called Chantodo's Elemental Seal. Elemental Seal is a black mage ability in Final Fantasy XI, and Shantotto is one of the most fearsome black mages of this universe. However, the orb was renamed with the 1.0.4 patch, becoming Chantodo's Force...
Azmodan's hellish domain is reminiscent of a realm of Oblivion from The Elder Scrolls IV: Oblivion, from the creepy bloodgrass to the towers lined with demon skin. The only new things are the giant demons that hold up the towers, but even they are reminiscent of Mehrunes Dagon.
Possibly a coincidence or an obvious play on words, but there's an achievement called Imp Slapped.
Also a reference to the game Dungeon Keeper, where you could slap your Imp workers.
One of the possible weapons you can find is the Ancient Bonesaber of Zumakalis. Sadly, it's unknown is the Remonster of the Pultonians appear to use it on.
Adria was bound to attract more than her share of "witch" jokes in the boss achievements, but "Really Most Sincerely Dead", which you get for killing her in Hardcore Torment mode without crowd-controlling enemies, takes the cake.
One fairly common name for an Elite monster is "Zuul".
In the Reaper of Souls expansion, you can find a legendary potion called "Bottomless Potion of Kulle-Aid", whose flavor text says "Oh, yeah!". Besides the usual healing effect, it allows you to burst through the walls created by elites with the Waller affix.
The Shenlong's Spirit set is a reference to the Shenlong myth from the Street Fighter games, complete with the following: both weapons resemble Ryu's gloves, one glove's flavor text mentions the existence of the myth itself, and the other glove references the quote that spawned the myth in the first place.
Flavor text of Eight-Demon Boots is quote of one Yolk Chen.