Busby worked for more than 30 years across many local radio stations, and more recently at BBC Sport and BBC Radio 5 Live.
He has been praised for his positivity and kindness in tributes on social media.
“This is such sad news,” BBC producer Sam Clark said. “The most positive, warm and encouraging person. Pleased that our paths crossed and devastated that he has been taken from us. Sometimes life just isn’t far.”
“Dom was a great broadcaster and a brilliant man. He loved sport and radio so had the perfect job,” said radio sports news editor Colin Patterson.
“You could hear the excitement in his voice when he was reading the sports news on air. He had a long and successful career at the BBC - 30 years at many of our local radio stations, and for the past 10 of those years he also worked for BBC Sport.
“He will be much missed by his colleagues and listeners to BBC radio. Our thoughts are with his family at this terrible time.”