The humble UrbanMech of BattleTech is widely regarded In-Universe as a death trap that exists solely to be a distraction until it is inevitably swatted out of the way by a proper battlemech. The Urbanmech is a 30 ton light battlemech that can't even keep up with an 100 ton assault mech like the Atlas, only carries enough autocannon ammo for about two minutes of combat backed up with only one Small Laser as an Ranged Emergency Weapon, and it lacks arm actuators meaning that it cannot throw punches and struggles to right itself if fallen over. They tend to explode violently when they appear in the BattleTech Expanded Universe. On the other hand, its autocannon allows it to deal significant damage to larger mechs and it carries a Jump Jet Pack to zip around in its ideal environment that it gets its name from.
Games Workshop, in the first of many entries, gives us Blood Bowl. Halflings compete with Ogres as the worst team in the game and they're certainly treated this way in the game fluff. Every halfling game mentioned in the rulebook usually results in the halfling team getting downright slaughtered (not as in they lose the game but as in most of their players are dead or horrifically wounded by the end of the game), and in one instance they lost to a team that didn't show up. They're halflings, though, so it's hilarious.
The Buffy and AngelRPGs have a Butt Monkey trait in the form of the three-point Outcast drawback. Such characters have a -2 penalty to all Influence rolls, and automatically attract the attention of any cruel or abusive character. The descriptive text is pretty much textbook Butt Monkey: "Because of their interests, charisma (lack thereof), habits or mannerisms, these people seem to have a permanent 'Kick Me' sign taped to their backs. They are to the movers and shakers what porta-potties are to construction workers."
In 3rd edition Dungeons & Dragons, due to his origin in Executive Meddling which also required him to be prominent in art, the very generic iconic Fighter Regdar is always seen getting killed in artwork.
Fusilier Angus appears in many examples of play in the Infinity free rules downloads. In most of them, he's being shot, falling off things, or otherwise having to make ARM rolls. The fluff for his special model in one of the Dire Foes packs is that coping with his incompetence has managed to make his partner, Fusilier Bipandra, an extremely competent field medic just from patching him up.
Though each different setting uses them in different ways, the Goblins of Magic: The Gathering are pretty much always some form of Butt Monkey. So much so that in a recent expansion, a goblin's greatest aspiration in life is to be eaten by a dragon.
Minotaurs, to the point that threeseparatearticles on Magic's website detail the various bad things that have happened to Minotaurs on cards over the years. What's more jarring is that "Minotaur" isn't a particularly common creature type in the game, so they appear on kill spells almost as often as the appear as cards in their own right.
Warhammer Fantasy has the Wood Elves. For the longest time, all you had was a PDF and some models that were difficult to find and harder to win with. Then they get an update and while they weren't too bad, they certainly were playable and a proper army. So why do they belong here you wonder? 8th edition. With a terrible magic lore (and only 1 Lord Choice that can access to others), overpriced and underpowered units and the vast majority of their units being skirmishers (which got hit hard under the 8th edition rules) Wood Elves are probably the weakest army in Warhammer Fantasy right now.
Beastmen can get this too, especially for older players. For a while, they were a solid army, which access to Warriors of Chaos to overcome their problems. But as the book got older and the other armies got updated and Beastmen weren't allowed to use them anymore, Beastmen became weaker and weaker, to the point where they fell into last place. Then they got their pre-8th edition update, which fixed some of those problems, but created whole new ones. They still function now, but they're a boring and lifeless army to play, which many players feel is too much like the Orcs and Goblins book, instead of a unique army on it's own. Wood Elves may be weaker but at least they're true to their playing style.
In the Yu-Gi-Oh! game, Goblin Attack Force appear to be the "designated stooge" of the game when a Spell or Trap Card needs on, as they are the butt of jokes on about 17 or so Spell and Trap Cards, being attacked or receiving a detrimental effect on most of them. This is likely a reference to the card's vulnerability after attacking.
An example of this trope played seriously is in Warhammer 40,000. The Imperial Guard tends to be Games Workshop's favorite punching bag as the guard tends to get picked on the most in other faction's battles in their respective codex. For a while, this extended to Imperial Guard players as well as they have had to play with an underpowered codex for 14 years spanning across 3 editions with half of their units being useless, and during 5th edition actually had a game type that was literately unwinnable because Annihilate was obviously not play tested with Imperial Guard. Forge World however sees the guard as The Favorite. IG players eventually got the bone tossed in a big way during the last few years as they got the mother load of Apocalypse releases and the new guard codex not only resolves the Annihilate problem but also allows guard players to use their Forge World goodies in normal games. Everyone else is scared that guard players are going to enter Who's Laughing Now? mode.
Even Relic treats IG players as the Butt Monkey. In Dawn of War, the guard is always the bottom tier faction. The only good thing about this is that if said guard player lives long enough, they get the Bane Blade. Who's dying now?
Examples of an old, painful joke regarding the IG - "Due to budget cuts, we've decided to replace your flak armour with cardboard and your lasguns with flashlights - the latter options being much cheaper and just about as effective. Remember! Faith is your shield, faith is your sword, just like it is for the Space Marines. What? No, no, the Power Armour and automatic 100cal gyrojet AP grenade launcher are just ceremonial...". And let's not get started on the Commissar jokes...
And for the Greenskins, there's the Grots, aka Goblins IN SPACE! Practically bullied around and used as slaves by the Orks, with very few actually serving on the battlefield as soldiers, and for a very good reason, since grots are extremely weak. Generally played for laughs.
Homunculi processed by the Afriel Strain process are blessed by inheriting the genes of the Imperium's greatest "heroes", possessing many skills related to the bloodline...they also have the absolute WORST luck...EVER. An example of this is Lorii, an Opposite-Sex Clone of Macharius who ended up in the Last Chancers for neglecting orders to save her brother, and got into her CO's bad side and was literally butchered alive.
From a purely developmental standpoint, the Dark Eldar have been Games Workshop's Butt Monkey for the better part of their existence. Although repeatedly promised a new codex (their current one is a slightly modified version of their original 3rd Edition release) and new models, which haven't been updated since their original release (and are widely considered the worst in the 40k lineup), they are still waiting for a long-overdue update. In the meantinme, pretty much every other race has received an update; with several Space Marine chapters receiving multiple. GW has claimed this is due to the small number of Dark Eldar players and poor sales of the line, which is most likely the result of the oudated and almost unplayable rules, and unappealing models.
Rectified as they have a new codex. They new models look amazing and while the Dark Eldar still aren't the easiest to play (do the words Glass Hammer mean anything to you?) they are definitely a workable and competitive army, especially in the right hands.
From an In-Universe example, the Iron Warriors during the Great Crusade. They were stereotyped into being the Crusade's go-to guys for seigebreaking and fortifications, and eventually that was all they were allowed to do, the Legion being stretched thinner and thinner as ever smaller bands of them were expected to conquer and control more worlds. Even their rival seigebreakers, the Imperial Fists, were given more honor and respect than the Iron Warriors. It eventually resulted in them becoming the most bitter of all the traitor legions.
Making things worse is that, according to the novels detailing the Horus Heresy, their Primarch, Perturabo, was actually a very sensitive and artistic soul who had never wanted anything more than to make beautiful buildings — all of his skills at designing fortifications and cracking defenses were side-effects of his love of crafting buildings. Even Fulgrim, by then fully corrupted to the worship of Slaanesh and eagerly pursuing daemonhood, seems genuinely to pity Pertaurbo when he finds out.