Disclaimer: yes, I realise there are a lot of common tropes in the fantasy genre as a whole, and just because Inheritance and a particular older work share one of these tropes, that doesn't necessarily mean Paolini took the idea from that work. I've included in this answer only those works whose similarities I believe are too great to be coincidence, or which Paolini has explicitly acknowledged as having referenced.
The Star Wars connection is perhaps the most obvious. That Wikia page goes into a huge amount of detail, so I'll just list some of the most important similarities and wind up with a pertinent video:
- Brom ~ Obi-Wan (first mentor figure, dies during first instalment)
- Main character turns out to be son of the Big Bad's chief lackey (it's actually more complicated in Inheritance, but still)
- Oromis ~ Yoda (second mentor figure, hidden away, his survival a secret, dies during thrid instalment)
- Rider swords ~ lightsabers (Eragon/Luke gets a sword/lightsaber early on, loses it, makes another)
- Rebellion vs Empire
- Riders ~ Jedi (formerly great organisation, now almost extinct, their last members turned evil; Eragon/Luke must rebuild them)
Lord of the Rings
Basically all works of fantasy copy at least something from Lord of the Rings; at this point it's hard to tell the difference between "copying an idea from Tolkien" and "using an idea that's standard in the fantasy genre". That said, here's a few points:
- Elves live in the forests; dwarves live in the mountains
- Elves are tall, elegant, and immortal; dwarves are short, bearded, and secretive
Dragonriders of Pern
It made me sad to discover that Paolini wasn't the first to develop the idea of human-dragon symbiosis in the way he did, and that this idea was taken from Anne McCaffrey's Pern series:
- A community of people set aside from society by the fact that they ride dragons
- Dragon Riders formerly revered but now much diminished, a shadow of their former selves
- A symbiotic 'bonding' relationship between human and dragon, taking place around the time the dragon hatches
- Main character's bonded dragon is (at least at first) the only fertile female dragon in existence
Paolini has freely admitted to being heavily influenced by David Eddings's Belgariad, Malloreon, Elenium, and Tamuli series. E.g. here:
Mr. Eddings’ The Belgariad series is a wonderful introduction to fantasy. A classic coming-of-age epic, it features a young farm boy with a mysterious past; a mad, twisted god for a villain; true love; thrilling duels and battles; a unique land; and some of the most interesting characters in the genre. Mr. Eddings influenced how I approach fantasy, both as a reader and as an author. The Belgariad will always have a place of honor on my bookshelves.
Jeremy Thatcher, Dragon Hatcher
Another book that Paolini has explicitly admitted to being strongly influenced/inspired by. See here:
Some of these volumes sparked the inspiration for the story that would become the Inheritance Cycle. One of the most influential of these is Jeremy Thatcher, Dragon Hatcher, by Bruce Coville, a tale in which a young boy purchases what he thinks is a stone, only to discover it is a dragon’s egg. Christopher explains:
I liked the idea so much, I couldn’t get it out of my head. So I asked three questions: what land would a dragon egg come from, who would find it, and—since dragon eggs can’t be common—who else would be looking for it? My quest to answer those questions led me to envision the story that became Eragon.
Wheel of Time
Only one direct reference I can think of here (as opposed to more general fantasy tropes shared by Inheritance, Wheel of Time, Lord of the Rings, and various others). It's a small thing, but similar enough that I'm sure it's deliberate.
- The Big Bad (Galbatorix/Shai'tan) had thirteen powerful followers known as the Forsworn/Forsaken.
Ursula le Guin
Some things in le Guin's writing are almost on par with Tolkien in terms of their far-reaching influence in other works of fantasy. In particular, her concept of "true names", which - although she wasn't the first to come up with it - is probably one of the main inspirations for this fantasy trope as a whole. It's worth mentioning, though, that Paolini explicitly mentioned her - as well as the folklore that preceded her - as an inspiration for his concept of "true names" (which, by the way, is one of the best versions of this idea I've come across):
The true names were one of my favorite things to write about. Fair mention to Ursula K. Le Guin and all the folklore I read growing up that introduced me to the concept.
Others (mentioned in a Reddit AMA)
I don't know exactly where these influences are visible in Paolini's finished work, but he cited them as inspirations in a very recent Reddit AMA:
Of course I was inspired by Star Wars! And also Lord of the Rings, and Dune, and Eddings, Fiest, Tad Williams, Anne McCaffrey, Beowulf etc. As with Lucas, I was drawn to the heroic monomyth, and I wanted to write my version of it, as a love letter to the genre.
(Does that mean Angela and Solembum are R2-D2 and C-3PO? And what about Elva? . . . Wait, I'm confused. :D)
And (from a different thread on the same AMA):
Sure, Star Wars was an inspiration. So too (and perhaps to a greater degree) was Raymond Feist, Tad Williams, Anne McCaffery, E. R. Eddison, Brian Jacques, Andre Norton, Jane Yolen, Frank Herbert, Ursula K. Le Guin, Mervyn Peake, and many, many more. It's all food for the imagination!
Thanks to @ibid for finding this hot-off-the-press news!
This reference has been explicitly confirmed by Paolini. About the following line from Arya in Brisingr:
"Adrift upon the sea of time, the lonely god wanders from shore to distant shore, upholding the laws of the stars above."
Paolini said in the afterword:
Also, for those who understood the reference to a 'lonely god' when Eragon and Arya are sitting around the campfire, my only excuse is that the Doctor can travel everywhere, even alternate realities. Hey, I'm a fan too!
It looks like Solembum also made a Doctor Who reference in the last book, but this isn't confirmed. See the following quote (emphasis mine):
"I have seen things that defy belief: whirlwinds of light spinning in caverns deep below the ground, men who age backward, stones that speak, and shadows that creep. Rooms bigger on the inside than the outside ... Galbatorix is not the only power in the world to be reckoned with, and he may not even be the strongest."
Paolini included a reference to the cult British TV series The Prisoner. For those who are fans of the show, it's a very clear reference; anyone else would never notice it at all. Here's Angela, near the start of the last book:
She raised her hand to her brow, thumb and forefinger touching in a circle, and, in an overly cheerful voice, said, “Be seeing you!” And with that, she sailed off.
These precise words and gesture is the standard greeting between residents of The Village in The Prisoner, and someone has kindly collated all instances into a Youtube video:
That's all I can come up with for the moment, but I may come back to edit this answer. For more details, you can see Criticism of the Inheritance Cycle on Wikia and the excellent, if scathing, blog post Eragon - Plagiarism Made Popular.