Is Emi a fanatic who takes her frustrations out on Maou because her violent Knight Templar tendencies don't fly in our world, or is her expectations of Maou perfectly justified after everything he did to her people, putting her in an impossible situation now that he's become harmless and sympathetic and her sense of justice won't allow her to just end the threat anymore?
Was Olba always evil, or did he find out the Church was planning to peg him as The Scapegoat no matter what, reasoned Then Let Me Be Evil, teamed up with Lucifer, and took it out on another world to express his frustration over being betrayed?
Way later in the Light Novels (#11 to be exact) we learn that Olba's journey to The Dark Side began early. He lost his faith seeing various contradictions in what the Church was doing. He came to the conclusion that the Angels of his bible were real Ancient Astronauts. He figured that the Holy Silver was some sort of technology or artifact but it wasn't until he met Lucifer that his ideas were proven true. His corruption was that he wanted to become a god, or, failing that, become one of the angels.
Awesomeness Withdrawal: Sadly, as in spite of great home video and light novel sales, as well as having a steady fanbase worldwide, there's still no second season in spite of the cliffhanger ending. Though White Fox only on rare occasions does second seasons.
Emi's Jerkass attitude towards Maou irks many viewers. Others find it hilarious and justified, given that he was pretty much fantasy Hitler in their world and is threatening to return and try to subjugate them all over again.
Episode 7 worsens her status. Is Emi punching and kneeing Maou hilarious or did it show how much of a bitch she is to Maou? Take note that Maou saved her from falling to the ground.
Is Chiho hilarious and heartwarming? Or is she annoying and controlling? Some fans think it's the first, while others think it's the latter.
Further complicating this matter is the sheer amount of Character Shilling she gets from the narrative, especially in the light novel. Even fans of the character concede that the narrative often bends itself backwards to tell the audience how amazing Chiho is, with characters frequently in otherwise unrelated conversations, noting how amazing she is, how strong her strength of character is, how kind she is, etc.
Is Alas Ramus an adorable character who adds new dimension to the story, or a rather oblique attempt to force Maou and Emi together? The addition of Acieth Alla, who is functionally the same character except aged up; further muddles this issue.
Big-Lipped Alligator Moment: In the anime's final episode, the villainous Sariel winds up in the MgRonalds supply closet after Maou tossed him into a portal to Ente Isla in the previous episode. He hits on Kisaki like a pervert, gets crushed under her heel, and then disappears from the story. The scene has nothing to do with the rest of the episode and pointlessly brings back a character whose role in the plot was resolved for two minutes of goofy sex farce. While the character in question does have a larger role in the books, their sole scene in this episode is an utterly superfluous narrative wrinkle with no payoff.
Broken Base: While even fans of this series agree the second half wasn't as good as the first half, it did improve on some things. Most notably the character interactions, development, and world building all while staying true to itself. These people often disagree with the notion that it went through a dip in quality. Others, however, were bothered by the joke quality decreasing, increased emphasis on the story and how it began to let light novel cliches sneak in. Though it is generally agreed that the show never got bad, and even people who don't care about it at least admit that it has merit.
Captain Obvious Reveal: The revelation that Miki Shiba's niece Amane is involved with Ente Isla would've been more surprising if her aunt hadn't already cryptically revealed she knows about it.
Draco in Leather Pants: Maou is this among both the fandom and in-universe. He's a nice guy who enjoys his job and just wants to make a living for himself and alongside of that fans seem to forget the most important detail about his character... he is actually the devil king responsible for the deaths of many people in Ente Isla. Mitigated in the light novels, which paint a much more complex picture of the war as they go on.
Ensemble Dark Horse: Some of the side characters like Emi's co-worker Rika, Maou's boss Kisaki, and Emi's servants Emeralda and Albert, are all reasonably popular for contributing entertaining bits whenever they show up.
Fandom Rivalry: With YuuShibu which was another light novel comedy that aired the same year involving demons and heroes in modern day Japan working part time jobs. Fans of The Devil is a Part-Timer often decry YuuShibu as a generic harem with a boring main character and having nothing to offer other than fanservice in spite of the clever jokes in this show. Not all fans of YuuShibu are critical of The Devil is a Part-Timer however a lot of them admit that most people are unfair on YuuShibu just because its a harem/ecchi show. Fans of YuuShibu will also point out how even though it is a by the books fanservice show it is at least honest with itself unlike The Devil is a Part-Timer which tries to work in romance elements and a more serious subplot.
Even Emerada needles her playfully about it, and she'd seen first-hand what Satan's army had done!
"Funny Aneurysm" Moment: Sarue creeping on Chiho and various women, when his English voice actor Scott Freeman actually would be convicted of child pornography.
Genius Bonus: In Judaic traditions, Sariel was a fallen Watcher angel who taught mortal men to use the lunar calendar. How did he fall from grace? Lusting after mortal women.
Harsher in Hindsight: In light of Scott Freeman's conviction as a sex criminal (as in: caught with Child Porn), scenes of Sarue creeping on various women is not as funny as originally intended and especially when he's creeping on Chiho who is a minor.
The Demon Lord and his most loyal general live together in a cramped, run-down apartment in which the former does the working and the latter does the housework. Oh my. Alciel actually calls himself a "house husband" in his bio, although he acts more like an overprotective parent to Maou at times. Lampshaded by Chiho's classmate who goes wild with yaoi fantasies upon learning Alciel and Maou are roomies.
When you throw Lucifer into the mix, who is a free loader who looks and acts like a teenager and combine that with the fact that Maou is the apartment's breadwinner and Alciel does all the housework, the household actually does look like a couple with their child.
Ship-to-Ship Combat: If you ship Maou with either Emi or Chiho expect fans of the other ship to disagree with you. And of course there are those who believe Maou X Alciel is the best pairing in the series and that both Emi and Chiho aren't suitable love interests.
Superlative Dubbing: In an age where many dubs of anime comedies are criticized for poor translation quality, this show did the opposite. As stated below the dub featured many re-translated jokes that still worked in the context of the show and made it just as funny to watch and even gave the show high rewatch value. It also helped that the cast (with the exception of Josh Grelle and Tia Ballard) mostly used newer voice actors to Funimation or voice actors with not a lot of experience in lead roles to give it a fresh feel. Though the veterans gave strong performances as well.
Woolseyism: The dub is full of local jokes not in the original, including many Shout Outs, natural language, and solid dubbing.