Tom Kratmann specifically points this out in A Desert Called Peace when he writes a paragraph that goes something like this: People's Republic means dictatorship. Democratic Republic means oppressive and corrupt dictatorship. People's Democratic Republic means really oppressive and corrupt dictatorship with genocidal ambitions.
In Foundation and Empire, the Republic of Korell, which is for all intents and purposes a one-man state ruled by an extremely repressive and vicious ruler named Asper Argo, who styles himself as Just the First Citizen and assures visitors that he is called the "Well-Beloved". It is also wretchedly poor, has a Secret Police and the infrastructure, like the population, seems poor and underdeveloped. The main character of the story, Hober Mallow, sourly notes that for such a beloved man, his house (which is more like a Palace) is unnaturally well-defended, heavily fortified and has a large complement of guards.
The Foundation itself becomes this by the end of the book, due to corruption and inefficiency, until it is shaken up by The Mule. The Mule himself sets up a Union of Worlds with himself as First Citizen of the Union — which is to say, he sets up an autocratic regime centred around him that only avoids some of the standard totalitarian pitfalls because the Mule can ensure loyalty in both active and passive ways.
Honor Harrington: In the backstory, the Republic of Haven becomes the People's Republic of Haven when it makes its legislature hereditary and becomes a Bread and Circuses welfare state, and keeps the name when it undergoes a revolution and becomes an unholy mix of Revolutionary France and Communist Russia. It only gets better after a second revolution, where it drops the "People's". The trope is discussed by William Alexander, Lord of the Manticoran Exchequer, when he complains that public opinion in the Solarian League backs Haven because it is a republic while Manticore is a kingdom, and the people of the Solarian League assume that a republic must be a democracy while a kingdom must be autocratic. Later on, Havenite Secretary of War Thomas Theisman muses that he really wishes he could just have Arnold Giancola, the guy he knows is behind the resumption of the shooting war with Manticore, taken behind a shed and shot, but specifically notes they have to do everything by the book to show they are not the People's Republic of Haven anymore.
In one of the Worlds of Honor short stories, a group of oppressed women and children escaping the planet Masada in a stolen spaceship need to figure out which ship in orbit they should seek help from. One from the Star Kingdom of Manticore, or one from the Peoples' Republic of Haven. They decide that they should avoid the latter, because if it really cared about people, they wouldn't feel the need to say so in their name.
Played with by the Star Empire of Manticore. They fit the dictionary definition (an aggregate of nations or people ruled over by an emperor or other powerful sovereign or government). But they aren't imperial, as in, they don't go out and conquer new territory. Every star nation that has joined has been the one to ask to be annexed. This is considered to be a superior fate to ending up under the aforementioned People's Republic of Haven, or the Solarian League, which is imperial in all but name.
In the Sword of Truth series, the fortress city of the royal line that rules D'Hara is called the People's Palace. The name actually fits, in a weird way: The entire structure's design is that of a power spell, meant to sap energy from spellcasters on the grounds and give it to the ruling Rahl. The spell form, though, is "drawn" with all the people moving through the palace, so without them, it would be powerless.
The D'Haran army in the first book was called the People's Army of Peace...
This has sort of happened in Wild Cards, with the People's Paradise of Africa, which encompasses the Congo.
Atlas Shrugged: Every non-United States country we hear about in Ayn Rand's novel is the "People's State of (Fill in the Blank)". Of course, they are all oppressive, poverty-stricken hellholes. Argentina and Chile are stated to have been non-People's States countries, and others are suggested to exist. The transitions seem both inevitable and to not go well. In Chile's case, the transition comes midway through the book, as just one more step the world is taking towards universal communism.
The Thursday Next series has a People's Republic of Wales which has a president for life and has opened a free trade zone in the manner of China. England itself in the series might count, as it is a republic rather than a monarchy, and although it isn't really markedly Socialist, it is fairly Orwellian and dominated by a Mega Corp..
Poul Anderson's "Withit's Collegiate Dictionary," from There Will Be Time, contains the following definition:
Republic: A country whose government is chosen not on a basis of heredity or riches but by the electorate, from whom political power grows. People's Republic: One in which the electorate consists of a gun barrel.
In Animal Farm, the Farm declares itself a republic years after it has turned into a totalitarian state, so that it can have one-candidate elections.
Nineteen Eighty-Four: Averted with Oceania, which is never given a pre-title at all. As O'Brien explains in the third-act Grand Inquisitor Scene, the Party has no illusions about what they are or what kind of society they are creating. Since "Ingsoc" is short for English Socialism, and Orwell himself was an avowed socialist, Oceania might have been at least nominally a republic at first, though the Ministry of Truth has since been ruthlessly eliminating all evidence that freer forms of government could ever have existed.
Space Captain Smith has two. As its name suggests, the Democratic Republic of New Eden is a hellish theocratic tyranny. Then there is the Greater Galactic Happiness, Friendship, and Co-operation Collective, which is run by demented sadistic lemming-men intent on conquest.
Inverted on the Discworld. Ephebe, the only democratic nation, is headed by a Tyrant. Yes, the Tyrant is regularly voted out of office.+ This also calls back to ancient Greece, when 'tyrant' a ruler via unconventional means such as populist coup, not inheritance or election, before it picked up its modern connotation.
Democracy is, in fact, considered inherently flawed by the rest of the Discworld, on the basis that there's no way to keep Nobby Nobbs out of the voting.
Played more or less straight in The Compleat Discworld Atlas, where it's revealed that the Agatean Empire is now the People's Beneficent Republic of Agatea. Madame Chairwoman is Twoflower's daughter Pretty Butterfly, who's not really evil as such, but the included list of regulations does suggest that the typical peasant doesn't have much more freedom than he did under a succession of insane emperors. And probably a lot less than he did under Emperor Cohen. This was foreshadowed back in Interesting Times, when Rincewind predicted that if the Red Army overthrew the empire and announced the guys standing in fields looking after water buffalo now ruled themselves "by means of the People's Committee" there probably wouldn't actually be many water-buffalo-string-holders on the Committee, and probably quite a few members of the Red Army.
In a Splinter Cell novel, Sam Fisher muses that, as a rule, the level of a country's dictatorship is directly proportional to the number of democratic descriptives in its name.
In the Skolian Empire series, the Eubeian Concord is The Empire, ruled by a race of sadistic anti-empaths with red eyes, called Aristos or Traders. Everyone else is a slave, and, as one character puts it, "the Aristos are only in concord about their desire to conquer and enslave everyone else, and their slaves have no choice but to be in concord with their masters."
In Shooting Script, by Gavin Lyall, the Republica Libra is a Central American state with a name that means (obviously) "Free Republic". Naturally, it's really a dictatorship run by whichever general or "liberator" won the most recent civil war, which one doesn't matter, and which lasts only until the next civil war.
Ape and Essence by Aldous Huxley has a post-apocalyptic Satanist theocracy which has to remind its subjects, when they want to do something the church disallows, that "this is a Democracy... in which every proletarian enjoys perfect freedom."
In Robert A. Heinlein's short story Coventry the most totalitarian nation in the Coventry is called "The Free State".
While The United Human Federation (which is a lie from the start since in only controls the inner planets) in Dani and Eytan Kollin's Unincorporated series starts out fairly democratic, in fact it starts by increasing the franchise by granting it to anyone who joins the military, it becomes less as the series progresses. By the final book a cabinet member admits that they are well on the way to becoming a socialist state and perhaps even a communist one, even though they started out, on the surface at least as a libertarian paradise.
In Timeline191, the CSA under Featherston. Firstly, despite its name, the Confederate States (which, in Real Life as well as in the series, started out as a confederation of states with a somewhat weaker central government), becomes a unitary totalitarian state, where everyone must fall in line with Featherston and take their marching orders from Richmond. Secondly, the Freedom Party is about anything but freedom, as is lampshaded in the series.
A Song of Ice and Fire had the old Valyrian Freehold, despite its name its very much The Empire, and it practiced slavery just as much as the old Ghiscari Empire it supplanted (though from the scant information we have, it seems to have been a matter of exact words in the same way the Roman Republic, one of the Freehold's inspirations: the Valyrian Freehold was a Freehold for Valyria itself... which was only a fraction of the territory under Valyrian dominion).
The Republic of Gilead, a Christian fundamentalist theocracy in The Handmaid's Tale. In Gilead (which is supposedly somewhere in the US), women have little to no rights, are forbidden to read, write or hold a job, and fertile ones are used as brood mares for the upper class. Society is rigidly divided, privacy is practically nonexistent, the secret police is everywhere, and rebels deviating from societal or religious norms are hanged publicly. It's also implied that Jews and people of color have been massacred.