Stanton’s agent John S Kelly confirmed the star had died on Friday afternoon at the Cedars-Sinai Medical Centre in Los Angeles in a statement.
Kelly said that Stanton, remembered by many as a “cult actor”, had died of natural causes.
Hamill posted a black and white photograph of Stanton on Twitter, writing alongside it: “Sad to say goodbye to the incomparable Harry Dean Stanton - so profoundly authentic in every role he inhabited”.
Lynch described him as a “great one” in a statement posted on the official Twin Peaks Twitter account.
He said: “The great Harry Dean Stanton has left us. There went a great one. There’s nobody like Harry Dean.
“Everyone loved him. And with good reason. He was a great actor (actually beyond great) - and a great human being - so great to be around him! You are really going to be missed Harry Dean! Loads of love to you wherever you are now!”
The pair had recently worked together as Stanton featured in Twin Peaks: The Return and they star together in the last project he worked on, the film Lucky, due to be released later this month.
A message posted on the film’s Twitter account said: “We lost a legend today with the passing of Harry Dean Stanton. Everyone involved with LUCKY is deeply saddened by this tremendous loss.
“We had always hoped to celebrate this film and Harry’s beautiful performance with Harry himself.”
“Our hearts go out to all who knew him, all who loved him, and his fans the world over. He will be missed, but his work will live on as long as people watch films.”
The film’s director, John Carroll Lynch, said he had “the honour to watch over Harry’s final performance in intimate, exquisite detail”.
In a long message posted on Twitter, he wrote: “In this nano-second world, I understand the need for a response to the death of Harry Dean Stanton. But it is difficult.
“I had the humbling good fortune and honour to watch over Harry’s final performance in intimate, exquisite detail.
“I watched every second of it again and again over hours, days, weeks and months these last two years. It was a gift beyond measure. He is amazing in it.
“But he was amazing in everything he ever did.
“And as a man, he was singular. Funny and mischievous, wise and furious, loving and kind, and always, always beautiful.
“I am sure there is more to be said, but for now, I celebrate his life, his work and will hug the loved ones he left behind.
“Thank you, Harry Dean. Thank you, Harry Dean. Thank you.”
During his career Stanton appeared in around 200 films and television shows, with his most memorable credits including Alien, 1984’s Paris, Texas, which featured Nastassja Kinski as his wife, and the TV show Big Love.
Big Love writer Dustin Lance Black posted a photograph of the series’ lead actor Bill Paxton and Stanton on Twitter, writing alongside it: “The most anxious I’ve ever been on set was the morning before working with #HarryDeanStanton. Now it all seems a dream #Legend #BigLove #RIP”.
He had also had roles in films such as the 80s coming of age romantic comedy, Pretty In Pink, in which he played the father of Molly Ringwald’s character.
Another notable performance of the Kentucky-born star remembered by fans was his role in The Godfather Part II.
Director and actor Samuel West tweeted: “Harry Dean Stanton in PARIS, TEXAS. He redefined acting for me. That someone could be silent, still and yet utterly mesmeric. RIP.”