Inspired by this thread. Frank Sinatra turned down dirty Hally Callahan, which became Clint Eastwood’s iconic role.
Robert Redford turned down the role of Benjamin Braddock in Th Graduate because he didn’t think he could portray his naivety. That led to Dustin Hoffman’s breakthrough role as the character.
When Robin Williams got the role as Garp, he suggested his friend Christopher Reeve for the transsexual Roberta Muldoon. John Lithgow was nominated for an Oscar for the role.
Something I did not know until today: Jack Lord had been the original choice to play James Kirk on Star Trek, but he wanted a 50% stake in the show, and was turned down. So, instead, the role made William Shatner famous. 
Leo turned down Boogie Nights for Titanic, inadvertently making Mark Wahlberg… somehow… a semi-respected actor for the next decade or so.
Sylvester Stallone and Mel Gibson both turned down The Terminator. James Healey turned down Mad Max.
Humphrey Bogart became a star taking roles George Raft turned down: High Sierra and The Maltese Falcon. Raft didn’t like the first because he died in the end, and the latter because he didn’t want to work with a first-time director.
Speaking of John Lithgow, he was the first choice to play Doc Brown in Back to the Future, but was not available.
Tom Selleck could have been Indiana Jones.
Christopher turned down the role of Roberta Muldoon because he felt pepole would not accept the actor who played Superman next playing a transsexual.
Has to be actively refusing the offer, right? Not unable to accept? Dougray Scott was cast as Logan/Wolverine in the first X-Men movie, but Mission Impossible 2 ran long. And of course we all know about Tom Selleck re: Indiana Jones.
Marilyn Monroe turned down the role of Holly Golightly, not wanting to play a call girl. Audrey Hepburn was already famous, but that sure is an iconic role.
Jennifer Aniston turned down the lead in Enchanted, either because of salary or not wanting to sing, depending on who you ask.
Eddie Murphy turned down the role of Winston in the first Ghostbusters movie.
Hugh Jackman was shortlisted for Casino Royale, but didn’t want to commit to another franchise.
Samuel L. Jackson turned down Don Cheadle’s role in Boogie Nights.
And while much is made of Molly Ringwald turning down Pretty Woman, the fact is that she turned down $3000, the original, darker version that ends with Edward pushing Vivian (if those were their original names) out of his car and throwing the money after her. Hardly a missed opportunity when that version never got made. And as far as her mother reading the Blue Velvet script and blowing a gasket, I’m not convinced that the role that went to Laura Dern was MR’s to turn down.
Mark Hamill did turn down the role of David Bradford, the eldest son in Eight is Enough, when the pilot he played the role in was picked up as a series. He had then committed to do Star Wars.
… thus making Grant Goodeve the beloved star of stage and cinema he is today… 
Hey! This was my 24,000th post!
Revision/additional info:
Apparently, Jeffrey Hunter (who had played the captain – Christopher Pike – in the first Star Trek pilot, which was later incorporated into the Trek episode “The Menagerie”) was the first actor approached about playing the captain in the second pilot, and turned it down. Stories vary as to what happened there – there are reports that he wanted to focus on movies, or that his wife was making unreasonable demands, or even that he was fired.
After that, Roddenberry apparently offered the role to Jack Lord, but wouldn’t agree to Lord’s demand for an ownership share, at which point, the role went to Shatner.
According to Mike Nichols’s commentary on The Graduate, Redford did want the role and asked him for it; they were friends and had worked together previously on the stage production of Barefoot in the Park. Nichols wanted someone who was more of a schlub–someone who would make you wonder what Elaine saw in him and why on earth Mrs. Robinson would want to seduce him. (No question about either of those with pretty young Robert Redford!)
To make his point, he asked, “Bob, have you ever been turned down by a girl?”
Redford: “What do you mean?”
Mickey Rooney was Norman Lear’s first choice to play Archie Bunker in All in the Family. He turned it down and Carroll O’Connor accepted the role that made him famous.
Arnold Schwarzenegger, Richard Gere, Clint Eastwood, Burt Reynolds, Sylvester Stallone, Harrison Ford, Don Johnson, Nick Nolte, Mel Gibson and Richard Dean Anderson all turned down the roll of John McClane in Die Hard - which made Bruce Willis into an action star.
Mel Gibson turned down the lead in Gladiator, much to Russell Crowe’s delight.
Sean Connery turned down the role of Gandalf in the* LOTR* movies, giving a big boost to Ian McKellen’s career.
Michelle Pfeiffer turned down the leads in both Silence of the Lambs and Basic Instinct.
Ditto Will Smith in The Matrix.
I don’t have my copy of Solow and Justman’s book handy, but as I recall, Hunter’s wife showed up for a screening of the first pilot and turned down the role for him, saying “This is not the kind of thing Jeffrey is interested in doing, and we feel it would hurt his career.”
Also, I believe Shatner got a piece of the show from the very beginning, as part of his signing deal. (IIRC, it was originally 5%, then went up to 10%; Nimoy eventually had a share as well.)
Shatner’s deal was kept a tightly guarded secret. When Nichelle Nichols learned about it years later, she reportedly said “That explains everything!”
And the weirdest part is, they were contractually obligated to offer the role to Frank Sinatra, and to wait years for him to relinquish the obligation.
After turning down Lord’s demand for 50%, Roddenberry must have decided that 5% was acceptable. 
I’ve never heard that before. Cite, please? No mention of it here, for instance: Star Trek Beyond: Who Got Really Rich from the Franchise | Fortune