Older Than Feudalism: The Ur Example of this trope is arguably Paris in the Iliad, who not only starts the 10-year Trojan War through which every character is slogging, but also behaves throughout as an effeminate, backbiting coward.
Older Than Steam: Journey to the West has The Load in the person of Xuanzang. He's a holy Buddhist monk, who is protected by four warriors. Not only he is both unable and unwilling to defend himself so he is easily captured. Because he is is a holy man eating his flesh grants immortality so there are a lot of demons who would want to capture him.
Ernie from the Grey Griffins book series. He's also kind of The Scrappy as well, with his nerdiness and whininess. But his dumbest move by far was when he just happened to randomly sneeze and blow out the match the heroes were using to light a dark room. Natalia was sufficiently pissed at him for it. Later on, he becomes more of a liability when he happens to end up in a coma caused by an enemy attack. Later, he is revived from the coma with magic powers, making him surprisingly powerful and no longer The Load.
Glew in book 5 of the Chronicles of Prydain, when he isn't The Millstone. The other characters are very aware of it, but keep him around because they feel sorry for him, have no way of sending him elsewhere, or want to keep an eye on him. To cap off his annoyance factor, he's constantly complaining that he's treated badly and the companions are being selfish in not thinking of his loss. He's so bad that this is actually his only saving grace in that book, when his greed leads to him inadvertently collapsing the mine tunnels that the group was traveling through, forcing them to take another route and leading them to accidentally find the missing sword and win the battle. Of course, he does redeem himself somewhat, in that he's able to confess to his mistakes at the end.
Kay in Greystone Valley winds up being this to an extent, since his spells get Sarah into more trouble than not. He does come in handy from time to time, albeit accidentally most of the time.
The Ciaphas Cain* HERO OF THE IMPERIUM novel Caves of Ice gave us a marvelous acknowledged example in the form of the tech-priest Logash, whose sole contribution is to tell them a little bit about ambulls before just hanging around. Some 70% of Cain's narration about him amounts to "What an annoying man he was" and/or "The mission would have gone smoother if I had thrown him to the orks/ambulls/necrons". However, he survived alone far better then any of the soldiers. In fact without him they wouldn't have escaped near the end, or been able to blow up the facility stopping the Necrons and saving millions of lives.
Eustace in The Voyage of the Dawn Treader until the point mid-story when he has his very own Anvilicious adventure in character development. He starts the voyage as a complaining selfish prig. The "Eustace's journal" portion of the book is a scarily effective portrait of self-delusion and self-righteousness.
Lex in the novel of Jurassic Park. Mostly justified as she is quite young and no matter how annoying she is, there is no way they are going to give up a little girl to vicious dinosaurs.
Amusingly subverted in John Maddox Roberts' Hannibal's Children and The Seven Hills, an Alternate History in which the Romans circa 100 BC are even moreBadass than in reality. Aulus Flaccus is lazy, hedonistic, and a poor fighter by Roman standards, definitely The Load — to them. By anyone else's standards, he's a terrifying killing machine. Attacked by assassins, he kills all four with six quick sword strokes... which his friend Scipio calls two too many.
Ce'Nedra. She becomes a liability to the heroes in Queen of Sorcery because after she runs away, everyone is looking for her because one, she's an Imperial Princess, and two, in the current political crisis, she's a very valuable potential hostage. She kicks up a stink about everything, treats Garion like a slave, makes it very clear to him that she thinks of him as a total inferior, and then gets pissed off that he won't totally devote himself to her. They have to leave her behind in Prolgu because she's the only one who can't defend herself at all, she treats everything as a personal insult, and despite her Crowning Moment of Awesome in bringing in the Tolnedrans and the Arends to join the army, it's balanced when she almost gets Adara killed by her own stupidity. In the sequel series this eventually becomes stated (by the Prophecies) as her official status: she's been messed with emotionally to the point where she can't be stashed anywhere out of the party's sight, and is there almost entirely to slow them down and prevent conclusion of the main plot until a variety of side tasks have been attended to.
Relg the Ulgo zealot spends quite some time weighing down the party by being terrified of the open sky, having to stop for prayer breaks several times a day, and being rude and obnoxious due to spending his whole life thinking himself special and blessed by his God, only to have his self-image crushed when said God personally tells him to get off his high horse, lay off his self-made holy quest and join the protagonists as he's supposed to. And as a zealot he also believes that women are out to corrupt him and that physical touch will ruin his spiritual purity. He grows out of it fairly quickly. Learning that praying on horseback is legitimate does help.
Rita Blakemoor in The Stand. A middle-aged socialite who has never known a real day of hardship in her life, she becomes The Load to LarryUnderwood, up until the point where she ODs on 'luudes and dies in her sleep.
Teela Brown is an innocent 20-year-old girl, part of the team that investigates the Ringworld, chosen because she was Born Lucky, literally and figuratively. The other members are a 200-year-old man who has done and seen everything, a Proud Warrior Race Guy and a Starfish Alien that's part of a system-spanning confederate. She's a liability from her sheer ignorance. Even more so once her luck starts kicking in, because it's her luck. It doesn't always translate into "lucky for the team". In the sequel, it becomes painfully apparent that her "luck" doesn't always even mean lucky for her personally; it's more "lucky for the human race in general."
Bella, from Twilight. She doesn't do anything remotely useful until Breaking Dawn and it's because of her bloodthirsty monsters keep showing up and attacking people in the surrounding areas. In the first book, James is attracted to her scent. In the second, so is Laurent. In the third, Victoria wants revenge for James. In the fourth, it's her hellspawn that causes the Volturi to come. Every major monster clash in all the books is a result of Bella.
Harry Flashman spends more time as The Load than anything else. He routinely deserts his men in the face of impossible odds, collapses into an uncooperative cowardly funk when his companions are plotting impossibly dangerous escapes, fakes or exagerates injuries before impossibly dangerous battles, and can be relied on to spill the beans and betray absolutely everyone the moment he gets captured. Being the quintessential Fake Ultimate Hero though, he always manages to make sure nobody ever hears about his less than heroic behavior. Flashman is 110% aware and unapologetic about this. Starting with his very first campaign he makes a habit, when events start turning south for the British Army, of collecting a force of tough fighters who can spirit his Loadishness to safety when disaster strikes. This keeps him alive and able to plan for silencing/hoodwinking/disposing of his saviors to preserve his reputation when the crisis is passed.
Special Circumstances: Barbara's husband, who does almost nothing but complain, particularly about having to actually take care of things for himself and the rest of the family when she's away on one of her missions.
Mommy of The Fire-Us Trilogy is a Hikikomori, which isn't a problem at first, since her job is pretty much to stay inside and take care of the little kids. However, it makes the later trip north out of Florida very difficult, and even once they figure out a way that she can bear being outside, she slows them down tremendously.
Peeta ends up becoming this in The Hunger Games after his leg gets injured. He's pretty much helpless, meaning Katniss has to risk her life twice as often to get food and supplies. Even after he heals enough to move around, he's a liability when Katniss fights and even when she hunts—he walks so clumsily that he scares off any prey within earshot.
He gets an undeserved bad rep for being this in Catching Fire as well. He displays these traits the first day of the Quarter Quell - at the Cornucopia (because he can't swim) and when they're fleeing from the poisonous gas (because he's weak after getting severely electrocuted mere hours before). However the rest of the time he's either fighting monkey mutts with Katniss and Finnick, carrying Beetee around the arena, creating a map of the clock, doing all he can to ensure that Katniss survives at the cost of his own life, killing Brutus who is one of their toughest competitors, or generally contributing as much as anyone else in the party. He was also the one who made sure himself, Katniss and Haymitch were prepared for the Quell by forcing the other two to train with him, both physically and survival skills-wise, and meticulously studying their competition to find out what the other victors' strength and weaknesses were.
On the other hand, Mags is literally The Load in Catching Fire, as she is an elderly stroke victim whom Finnick carries around in the arena. She was his mentor, and volunteered as tribute to save his love, Annie. She eventually gives up her own life by walking into a poisonous fog, so that Finnick will turn his efforts to protecting Katniss.
An interesting variant in the Codex Alera. Gaius Sextus is the First Lord of Alera, and probably the single most powerful furycrafter in the world. However, during the course of one of the books he has to get to the city of Kalare, held by the rebel High Lord Kalarus, in order to disarm Kalarus's Doomsday Device. The problem is, Kalarus has set up a network of sentinel furies such that if Gaius uses his powers anywhere within three hundred miles of Kalare, Kalarus will know and come down on him with a small army of furycrafters. So, Gaius and the two protagonists have to hike three hundred miles without Gaius ever using his crafting. Also, Gaius is over eighty years old, though the effects of age are partially mitigated by his watercrafting.