Takiya mentions Ant-Man in the English translation of chapter 42 when suggesting that Tohru could cure Kobayashi's illness with a "Fantastic Voyage" Plot.
In chapter 63, Ilulu does a happy version of the "Angry Dog Noise" face.
The comment on the magazine version of the title page of chapter 116 references the Street Fighter series by pointing out it's about a literal "ryu (dragon)" and "ken (sword)".
During "Time for Death Metal", Kanna, on a microphone, gives a Dragon Shout that sends Yana and Saikawa flying across the room. Also, the name of the page it's on is DragonForce.
In one chapter, Kanna and Saikawa are playing the Dragon Maid universe's version of Minecraft.
One of the games Fafnir comes across in Chapter 2 is Torneko's Great Adventure.
Anime
Season 1
The river seen in the opening credits looks remarkably like the very polluted river in the opening of the first episode of the OVA of Bubblegum Crisis. It's a lot cleaner looking in Dragon Maid though.
In episode 2, Tohru is shown holding a copy of Tsuru no Ongaeshi, which is a story about a crane that takes on a human form to serve an elderly couple after they save her life. She even tells Kobayashi not to come into the kitchen while she makes dinner as a reference to the fact that the crane told the couple to never enter her room as she made cloth for them to sell.
In the dub, Kobayashi quips this after flicking treats for Kanna...
Tohru: Fafnir does have a slight dislike for humans. We're talking about an ancient cursed dragon that has been guarding his treasure in a cave for, like, ever.
Fafnir and Makoto are seen playing a Dark Souls expy during the party in Episode 3. At the end of the episode, Kanna plays something that bears a suspicious resemblance to Mario Kart.
In episode 6, Kanna plays the suspiciously Mario Kart-like game at Saikawa's house, and Fafnir and Makoto decide who gets to wash the dishes by playing the Spy vs. Spy video game.
In episode 9, when Shouta reveals what was written on his slip during the scavenger hunt portion of the sports festival, a coin grabbing sound effect from the Super Mario Bros. series plays.
In Episode 11, when once again Tohru is trying to get Kobayashi to eat her tail meat, she says in a Green Eggs and Ham-style, "Could you? Would you for the new year?" to which Kobayashi replies, "I could not. Would not."
During the "Miss Kobayashi's RPG Dragon" segment in the fourth DVD special, Tohru, Elma, Fafnir and Makoto take part in a Final Fantasy battle.
Season 2
The opening starts off with shots of zooming on the various characters in what appears to be a Homage Shot to the first opening of Nichijou.
Tohru's realization that she's working as a chef and not a maid at the maid cafe has her briefly imitate Edvard Munch's "The Scream".
Episode 4: The "plasma ball wisp" waves of psychic rage Tohru appears to be emitting when telling the punks to get lost for the second time, are identical to the psychic pressure visible among Newtypes in the UC Gundam series (particularly Hamarn's in Double Zeta). It's overlayed with more modern "flame" effects, possibly to minimize the IP issues.
" (link to an SFW article on a NSFW site) at the end of chapter 35.