He died peacefully at home, his agent said.
Versatile and prolific, he was known and loved for his hangdog expression, lugubrious delivery and the often testy demeanour he gave to his characters.
As Time Goes By saw him star with Dame Judi Dench, a partnership they revived in Bond film Tomorrow Never Dies.
He also acted in Mrs Brown, again with Dench, and The Madness of King George.
Dame Judi, who starred in nine series of As Time Goes By with Palmer, told BBC Radio 4’s Front Row programme: “Geoffrey was master of comedy, an absolute master.”
Paying tribute to his “wonderful deadpan expression”, she added: “I’ve admired him all my life. How lucky to have been in something with him for so long.”
His co-star in Butterflies from 1978 to 1983, Wendy Craig, told the programme: “He was just a delight to work with, his timing was perfect.”
Despite his “rather serious face”, she said he was “full of fun” in person. “When he laughed and when he smiled his whole face lit up, his eyes twinkled. He was always up for a laugh and not a heavy-going serious person at all,” she said.
His early television roles included appearances in The Army Game, The Saint and The Avengers and he went on to appear in Doctor Who and the Kipper and the Corpse episode of Fawlty Towers.