Cox, 57, wrote: "The size of gratitude you brought into the room and showed every day on set is the size of the gratitude I hold for having known you. Rest In peace James."
Kevin S. Bright, Friends executive producer, said: "James Michael Tyler, our Gunther, passed away last night. He was an incredible person who spent his final days helping others. God bless you James, Gunther lives forever."
A post on the official Friends Twitter page read: "Warner Bros Television mourns the loss of James Michael Tyler, a beloved actor and integral part of our FRIENDS family. Our thoughts are with his family, friends, colleagues and fans."
His manager said: "If you met him once you made a friend for life. [He] is survived by his wife, Jennifer Carno, the love of his life.
"Wanting to help as many people as possible, he bravely shared his story and became a campaigner for those with a prostate to get a... blood test as early as 40 years old."
Tyler's Friends alter ego worked as a waiter and manager in New York's Central Perk coffee house and remains a popular figure among fans of the comedy, despite not having as much screen time as the six main cast members.
His character had a crush on Rachel throughout the course of the show, which ran from 1994 to 2004.
Tyler was absent from the Friends reunion earlier this year and explained that he only appeared via Zoom as he didn't want to put a "downer" on the show by having to explain his illness.
He said: "I wanted to be a part of that, and initially I was going to be on the stage, at least, with them, and be able to take part in all the festivities.
"It was bittersweet, honestly. I was very happy to be included. It was my decision not to be a part of that physically and make an appearance on Zoom, basically, because I didn't wanna bring a downer on it, you know? ... I didn't want to be like, 'Oh, and by the way, Gunther has cancer.'"
