Kenzo Tenma, brain surgeon supreme, is one of these in Monster.
Lelouch and Nunnally from Code Geass, after the former falsely claims their deaths in the Britannian/Japanese war that resulted in the colonization of the latter, seeking refuge with the Ashfords to avoid being used once again as political pawns by the Britannian Empire. In R2, Nunnally is readopted into the family (as a pawn against Lelouch), and Lelouch is left alone.
Anak Zahard's mother in Tower of God, a former Princess of Zahard who broke the law by getting pregnant and "uncontrolledly spread the power of Zahard" that way.
Princess Arika and Princess Theodora of Mahou Sensei Negima! during their time with Ala Rubra, and thanks to later events and revelations revealed after the Ostia Tournament, we could add Arika's son, Negi here as well.
King Van of Fanelia from The Vision of Escaflowne spends most of the series on the run from the Zaibach empire, who need him so they take control of the Escaflowne themselves. While the royalty of countries like Austuria are at least somewhat sympathetic to his plight, they're not entirely willing to risk their asses by truly granting him refuge.
At the conclusion of the Avatar Islands Arc, Empress Mercury is forced to retreat and abandon her holdings, takeing her forces into exile. This has severe political implications, as she is a Keeper with a reputation (possibly the most powerful one on the planet at that point) and the only reason she was forced to retreat is because threeDark Gods teamed up against her.
Sothe and Micaiah were these in The Tainted Grimoire after they realised that Vaticus Finch was the one who killed their unborn child and was trying to seize power.
Princess Leia in Star Wars spent years in hiding, running away from the Empire. This is a subverted trope in that she was already a member of the Rebellion before she went into hiding and going to them was logical after being captured at the start of A New Hope, possibly even if Alderaan had not gone kablooey. In fact, that may have been where she was headed even before Vader thought it'd be fun to sack the Tantive IV.
Her mother and adoptive father also fit this trope by the end of Revenge of the Sith. The Emperor was in control of the Senate, forcing them to go into hiding. Of course, Amidala tragically did not survive long.
Dr. Kimble - The Fugitive himself, who stops several times to help with medical emergencies, even if it means blowing his cover.
Subverted with Maladict in the Terry Pratchett novel Monstrous Regiment. When Maladict joins the army, the recruiting officers are wary of what would make a gentleman sign up, and Maladict immediately assures them there isn't a price on his head (before they get round to asking).
The Silmarillion is full of these. Beren is probably the cleanest-cut example; his father's kingdom gets overrun by Morgoth's army, and he lives off the land rather than leave it, first as part of a group of twelve warriors, then alone after the group gets betrayed. There's also Túrin, Elwing, Eärendil, and probably a bunch of others that I'm forgetting about at the moment.
Still on Tolkein, The Hobbit features both Thorin and Bard, whose ancestral kingdoms were laid waste by Smaug.
Paul Atreides and his mother in the original Dune seek refuge among the Fremen after the Harkonnens kill his father the Duke Leto. Whom he rallies into an unstoppable army and uses to claim the Golden Lion Throne. In the prequel novels, the entire House Vernius goes renegade after their world of Ix gets invaded by the Tleilaxu with the secret support from Padishah-Emperor Elrood IX and, later, his son Shaddam IV. All but one end up dead before Rhombur Vernius re-takes the planet with the help of the Atreides and receives his father's title of Earl.
In Sergey Lukyanenko's Seekers of the Sky duology, Junior Prince Marcus, a bastard son of the Possessor, is hunted by the entire State and, later, by the Russian Khanate as well.
Sort of happens to David in The Bible. He was just a shepherd boy, but was anointed as the king of Israel by the prophet Samuel. Unfortunately, Saul was still the king, and was not very excited about this. David eventually ends up on the run from Saul.
A Song of Ice and Fire has lots of examples. Viserys and Daenerys Targaryen became fugitives after their father, King Aerys, was overthrown in a rebellion and they escaped to the Free Cities.
Arya Stark is on the run to escape the Lannisters while her sister Sansa is held hostage at King's Landing. Ser Dontos Hollard, whose life she saved by asking Joffrey to make him the court jester, organizes to have her spirited away by boat right into the arms of Littlefinger, who helps her escape to The Eyrie.
When Theon Greyjoy betrays Robb Stark and captures Winterfell in the name of the Iron Islands, he makes Bran proclaim Theon as Winterfell's ruler. When Theon is betrayed by Roose Bolton, Bran, Rickon, Osha, and Hodor escape Bolton's sacking by hiding in the crypts while Theon pretends to kill Bran and Rickon by murdering two other boys who looks like them to "set an example". The rest of Westeros believes them to be dead as they are joined by Jojen and Meera Reed to go North and find the Children of the Forest to help Bran with his greendreams and learn to control his warg abilities.
Later, Tyrion Lannister, after being accused of killing King Joffrey, kills Tywin Lannister, his father and then runs away to the Free Cities with the help of Varys.
In the Belisarius Series, Princess/Empress Shakuntala, after she is rescued from Venandakatra the Vile.
Doremus becomes this at the end of It Can't Happen Here. He returns to America to aid dissidents as part of the resistance against the totalitarian regime. He's constantly on the run because Corpos are hunting him, however.
Odysseus sort of runs into this situation; though he isn't directly ousted, his prolonged absence causes his realm to be taken over and plundered by his wife's villainous suitors, and when he finally returns from his various voyages, to reclaim his land he needs to disguise himself as a beggar, gather his allies, and fight a battle against a small army of noblemen.
Robin Hood, in those stories that make him the Earl of Huntingdon or Robert/Robin of Locksley.
Legends of Huineng, the Sixth Patriarch of Zen Buddhism, state that after being secretly elevated to the position in the middle of the night by the Fifth Patriarch, he spent sixteen years on the run from disciples of Shenxiu (the head monk who everyone had assumed would become the next patriarch).
Many nobles in the Fire Emblem series become disenfranchised and hunted before either joining or forming a massive army to reclaim their kingdoms, starting from the original Marth, and later reflected by supporting Characters in the Tellius universe in Elincia of Crimea and later Sanaki of Begnion.
Gorath from Betrayal at Krondor, who turns out to have been Delekhan's rival for the throne (though he refused to compete for it) before his exile.
In Dragon Age: Origins, the Player Character of the human noble origin goes rogue. This is actually because he/she is a Grey Warden, rather than because he/she is the last (known) surviving member of the Cousland family.
The dwarf noble is in the same situation, having been exiled from the dwarf kingdom for murdering his/her older brother (either having actually done it or having been framed for it by the third sibling, depending on how the player decides to act).
In the prequel novel The Stolen Throne, Moira Theirin the Rebel Queen is the rightful ruler of Ferelden, whose father was forcibly removed from his throne by an Orlesian usurper. Since then, she has formed a rebellion against the occupation. After she is assassinated, her young son Maric becomes the King-in-exile. There are also several nobles that join the rebellion.
In Fable III the main character is pretty much this right from the get-go.
Ramza Beoulve of Final Fantasy Tactics was a member of the noble Beoulve family; as a knight turned mercenary, he eventually defected upon discovering that his brother killed his father and started to orchestrate a rebellion to gain the throne. For the remainder of the game, he remains a fugitive, trying to avoid capture by the kingdom and the Church.
Raphael Sorel of the Soul Series. He first met the little orphan girl Amy when he was on the run and hiding from the authorities. Amy pointed them in the wrong direction since she disliked them. Raphael was so grateful to her that he took her in as a ward and devoted himself to creating a perfect world for her. Then he learned about Soul Edge, believed like every other idiot who ever sought it that it was the answer to his prayers, and things went downhill from there.
Bonny Prince Charlie. When he was beaten at the Battle of Culloden, he was on the run from English forces for some time before escaping to France. During his flight a Ł30,000 bounty, a veritable fortune, was offered for his capture, and though many certainly had the opportunity (as he received aid from a number of supporters while a fugitive), none betrayed him.
We watched thee in the gloamin hour We watched thee in the mornin grey Though thirty thousand pounds they gie, Oh, there is nane that would betray!
White Russian exiles from the Russian Civil War played a part in politics in the 1930s and 1940's. Many were merely mercenaries or spies, but the most notable of them was Field Marshal Mannerheim. While not all of these were hunted some were, or had reason to fear that they were.
Harald Hadrada. He had to flee Norway after his brother, the king, was killed in battle. After some time in Russia, he worked as a mercenary for the Byzantine Emperor, soon becoming a commander. After learning that his nephew had taken control of the country, he used his newfound wealth to raise an army and force his nephew to accept him as co-ruler. A couple years later, the nephew died, leaving him sole ruler.
Henry Tudor spent almost all of his early life as either a fugitive or a glorified hostage, being the House of Lancaster's one last hope of returning to power after the rest of the line was exterminated.