For many Father Ted fans, Laffan will always be remembered for his beloved role as the milkman Pat Mustard.
Through his career he appeared in nearly 40 films and made 30 television appearances.
The Gate also tweeted a tribute, describing Laffan as "an incredible force in the Irish theatre community".
Father Ted creator Graham Linehan was among the first to pay tribute to the late actor. "We send our heartfelt condolences to his family and friends".
The Irish actor, who played Pat Mustard in the sitcom, has died aged 79.
"While Pat is perhaps still best remembered by Irish screen audiences for his portrayal of Mr Burgess in Roddy Doyle's film The Snapper (1993) and as Pat Mustard, the notorious milk man in Father Ted (Channel 4, 1998), he had nearly 40 film credits and 30 television credits to his name", the statement continued. "His career at the Abbey started in 1961 and spanned five decades".
A third penned: "Sorry to hear Irish actor Pat Laffan passes away.
He will be sorely missed".
Laffan also had the role of director at the Peacock Theatre, and directed at the Gate Theatre between 1979 and 1982.
The Gaiety School of Acting, of which Laffan was a board member, expressed their sympathies to his family and friends, while thanking him for his work with them over the years.
Love/Hate actor Laurence Kinlan posted a black and white headshot of the actor and tweeted: "I'm devastated to hear of the passing of the awesome Pat Laffan".
They gather enough evidence to get him fired, and Dougal takes over the milk round, but Mustard gets his revenge by rigging the float to explode if it falls below 4mph. He was one amusing man under his hard exterior.
He said: "Very sad to hear of the passing of Pat Laffan".
Many tweeted about the news on Thursday evening, with one posting: 'RIP Pat Laffan, from The Snapper to Father Ted, the provider of some of the biggest laughs we've shared as a nation'. In the space of one short episode, Pat Mustard became one of the most memorable characters in sitcom history.
Throughout his career, Laffan appeared in nearly 40 films and made 30 TV appearances, including in BBC's EastEnders and in RTE's The Clinic.