The Tourist class is basically one big Shout-Out to Discworld. The quest leader's name is even Twoflower!
The Archeologist class uses a leather jacket, a fedora and a bullwhip, just like a certain other famous archaeologist.
Many of the game mechanics are shout outs to CRPGs that were famous when NetHack first came out, but are obscure or even forgotten today. For instance:
'K'icking walls to open secret doors is a shoutout to the early games in the Wizardry series.
The ritual of using a magical bell, book and candle to open the way to the endgame is straight out of Ultima IV (which, in turn, may have borrowed it from Zork.)
The Wizard and Amulet of Yendor are named for the ultra-obscure... Amulet of Yendor.
As you might expect from a kitchen-sink fantasy game, there are numerous shout-outs to The Lord of the Rings and The Hobbit, including the magic inscription 'Elbereth', and the sword Orcrist and the dagger Sting. #chat-ting to a peaceful hobbit will prompt them to ask "about the One Ring". They also have a small chance of spawning with a dwarvish cloak or a elven mithril coat.
The gold coins are called zorkmids. This was the standard unit of currency in Zork.
The species names for the coyote monster are based off of Wile E. Coyote and the Road Runner. If the coyote is cancelled, the name becomes its real-life nomenclature.
A raven will respond "Nevermore!" should the player character chat with one.
As of Nethack 3.6.0, there is a 33% chance of releasing "the picture-painting demon" upon breaking a camera, in reference to how cameras work in Discworld.
Some of the things a T-shirt can read when read are:
Mail generated through a wish or in a bones file will be addressed to Larn, a video game that sends you junk mail if you win.
#chat-ting with a gecko while hallucinating might prompt it to tell the player "15 minutes could save you 15 zorkmids", in reference to GEICO insurance.
#chat-ting with Death has a 1/10 chance of catching him reading a copy of Sandman #8, where another character named Death is introduced.
Novels, added in v. 3.6.0, have names that reference Discworld.
Credit cards can be read in v. 3.6.0, and some of the possible banking institutions are from other media:
"Ankh-Morpork Merchant's Guild Barter Card" and "Ankh-Morpork Thieves' Guild Unlimited Transaction Card" are both from Discworld.
There are credit cards from banks from Larn and Omega, fellow Roguelike games.
adds a ton of additional messages, including to stuff as diverse as Undertale, Monty Python, and Phineas and Ferb. One False Rumor claims that "TVTropes are not addicting at all.".