Mr Rees, who was born in Aberystwyth, Ceredigion, won Tony and Olivier awards in 1982 for his title role in the The Life and Adventures of Nicholas Nickleby.
He played millionaire Robin Colcord in 1980s comedy Cheers and Lord John Marbury in US political drama The West Wing from 2000 to 2005.
Mr Rees had lived in America for 25 years and died after a short illness.
He played a love interest to Cheers' Rebecca Howe, played by Kirstie Alley.
She tweeted: "He is the sweetest man in the world! Lucky to have had him as a friend and to have worked beside such greatness."
His other credits include an appearance as the Sheriff of Rottingham in Mel Brooks' slapstick comedy Robin Hood: Men in Tights and Sherlock Holmes in a 1988 adaptation of The Hound of the Baskervilles.
Actor and director Samuel West tweeted: "Woke to the sad news of Roger Rees's death".
Choreographer Matthew Bourne wrote on Twitter: "RIP #RogerRees - a lovely, generous & kind man & an heroic & passionate actor. We all fell in love with him in #NicholasNickleby so sad."
Musician Rick Wakeman added: "Very sad to learn of the passing of Roger Rees. I saw him last year at a Water Rats Lodge. A truly lovely man and very talented too."
Mr Rees spent most of his childhood in south London where he acted in church and Boy Scout stage productions before studying art and painting.
His first job in show business was as a scenery painter before he turned to acting full-time in the mid 1960s.
Mr Rees's stage roles included playing Gomez Addams in a 2011 musical of The Addams Family and starring as Vladimir alongside Sir Ian McKellen in Waiting for Godot.
He lived in New York City with husband, playwright Rick Elice, whom he had been in a relationship with since 1982.
Welsh actor Roger Rees, star of Cheers and The West Wing, has died aged 71.