World of Warcraft, being an open ended MMORPG, is not subject to unwinnable situations as a whole (it's more of an Endless Game). But there have been numerous occasions when specific quests or dungeons have been broken because of scripting failures or other issues. In most cases, these have to be fixed by a game patch. Notorious ones:
Many quests involving scripted NPCs could break if the script failed to leave the NPC in an interactive state upon completion. This prevents any other players from getting or completing its quest(s) until the server is restarted or a kind GM despawns the offending NPC.
One common example is one quest in Gilneas, the Worgen starting zone, when, as you're sent to retrieve a cat, you encounter a Forsaken spy, but Grandma Wahl, in her worgen form, arrives and helps you take down the spy. However, sometimes, the spy doesn't die, leaving you and any other Worgen players at that point unable to proceed.
Nefarian, Final Boss of the Blackwing Lair raid dungeon, despawns temporarily if the players fail to kill him. But there was a bug where the door separating his room from the rest of the dungeon, which locks during the encounter, would fail to open again, shutting raiders out unless they had wipe prevention or left the dungeon entirely for 30 minutes to allow it to reset.
Razorgore, the first boss of Blackwing Lair has a unique mechanic in which one player is required to use an orb to mind control him. The orb has a set time limit to use and when it expires it must be used again by that player or any other. A bug would cause the orb to become unusable to any player resulting in Razorgore running to the orb and exploding, killing the entire raid. This bug was only fixed by the time of Cataclysm years later, where the orb became consistently usable and Razorgore no longer explodes when reaching the orb.
Scourgelord Tyrannus, last boss of the Pit of Saron dungeon, is supposed to dismount from his undead dragon Rimefang at the start of the battle and fight you on foot while Rimefang bombards the group with ice blasts from above. Sometimes, though, he dismounts and then immediately remounts again, and then spends the entire battle on Rimefang, out of range and unattackable.
The workaround: make sure no one attacks him until he begins attacking you. He should automatically aggro a few seconds after landing; then the party is free to engage him. The problem is making sure none of your party members attack him before he's fully ready.
During Wrath of the Lich King, the Death Knight starting quest chain could become unwinnable if you died early on and couldn't get back home. This would prevent you from learning key class abilities and getting your talent points. Fortunately, there was little to lose by deleting the Death Knight and rolling a new one. In Cataclysm, you no longer have to finish specific quests to get your class powers.
During the Alliance "retaking Gnomeregan" event, if anyone headed into the tunnel before the gnome leader, the event would bug out. The problem? He yells "Into the tunnel!!" to all the players present before beginning a scripted battle outside the tunnel.
The Battle of Darrowshire is known for two things: concluding an epic Tear Jerker storyline and being prone to bug. The player is supposed to use a quest item to summon ghosts to re-enact a battle, and then make sure one particular ghost isn't "killed" before his time. Sometimes the ghosts would simply refuse to appear. Other times Captain Redpath, the ghost you're supposed to keep alive, would simply drop off the face of Azeroth, and the battle would go on indefinitely, preventing anyone else from doing that quest until server reset.
Mutanus the Devourer, the final boss of the low level dungeon Wailing Caverns, appears after two waves of less intimidating monsters. Except sometimes you deal with the waves and are left standing there, staring into the waters from which he's supposed to emerge. If that happens, you've no choice but to leave and either let it be or start all over again. This is caused by attacking and killing the waves while they're still in the water.
Wailing Caverns in general may be the buggiest instance in the history of the game. For a long time, the place could be nearly unwinnable at-level due to glitches that caused enemies to spawn inside walls or underground (meaning you couldn't attack them because they were out of your line of sight, but they could still attack you) and others to disappear completely mid-stride and then reappear at the most inopportune moments. Since the backstory is that it was corrupted by the horrible visions of a druid trapped in the Emerald Nightmare, such twisting of the game's reality is fitting; but that's not much of a consolation when one of those blasted wall-druids has just killed your healer.
Not one, but both of the first two end-raid bosses in The Burning Crusade expansion pack were Unwinnable By Mistake until the 2.0.10 patch.
The naga queen, Lady Vashj, final boss of the Serpentshrine Cavern raid was so badly bugged that even when she was finally defeated for the first time, she instantly respawned and killed the entire raid. One of her most frustrating bugs involved her Mind Control ability that would affect 3 random targets. It shouldn't have been able to choose her current tank as a target... but it could, making her totally unbeatable unless your were lucky and your tank was never picked as a target.
The mad Blood Elf, Prince Kael'Thas Sunstrider, final boss was of the Tempest Keep raid was Unwinnable By Mistake because the mechanics of his encounter involve him being un-attackable until you've defeated several other challenges. Before being fixed, his aggro list (the priority order that determines who he attacks) would get filled up with all the healing that was being done to raiders during the first part of the fight. When he finally became active, he would attack and kill all the healers while tanks futily assaulted him, failing to get his attention as they tried to catch up with several hundred thousand points of healing aggro.
Some fans have speculated this was more Unwinnable by Design. Each expansion has tiers of gear. 2.0.10 was still in the first "season" of TBC and said bosses dropped gear for the second tier of the expansion. Around that one can note a sudden drop in "Unwinnable by Design" bosses and new bosses having other pacing mechanisms added that slow beating said bosses.
C'Thun may have been an example of this. Raiders had been wiping on him without success for several months, with the first kill coming within a day of a hotfix. Some contend that with superior gear he would have been defeatable (no kill till patch), since one key issue (additional opponents spawning in an area where they weren't supposed to and the damage was supposed to be primarily environmental - claws in C'Thuns stomach for the initiated) continued, along with all characters being killed should none be present outside of C'Thun's stomach. The latter issue still causes problems for players that outlevel and outgear the instance, since environmental damage scaled up (reportedly with spellpower) so higher level characters were still killed very rapidly in the stomach phase and eventually the only remaining characters are sent to the stomach, causing an instant wipe.
C'Thun's most infamous bug, though, was definitely a mistake. He has a chain-lightning style attack that bounces between players, dealing increased damage with each one it hits, to force players to spread out properly. However, he would sometimes cast it on raids as they entered the room, instantly killing most of the group.
On the first few months of Cataclysm, the final quest of Vashiir was prone to this. To avoid spoilers, a large battle happens, and is supposed to go into cutscene mode. However, sometimes it didn't, letting your character wander about and attack people. If you attacked even one person, the semi-cutscene would break, and the quest would not be flagged as completed, so you would have to restart the quest with no idea of what you did wrong.
Vanessa VanCleef, last boss of Heroic Deadmines, sometimes despawns if her aggro target falls off the top deck.
The 10-man version Horridon in the Throne of Thunder raid was nearly impossible to beat when the instance was first implemented. Somehow, the programmers forgot to scale the HP of the adds to 10-man difficulty, leaving them with the same HP as their 25-man counterparts. This meant it was mathematically impossible for the average 10-man group to defeat Horridon.