Dwyer came close to winning a Brownlow Medal on a few occasions, finishing second in 1961 and again in 1967 as well as placing third in 1960. He was a Syd Barker Medalist in 1961 and 1967, and was selected on the wing in North Melbourne's official 'Team of the Century'. Many supporters nicknamed Dwyer as "twinkletoes" because of his prowess as a ballroom dancer. Dwyer was part of the 1975 Premiership as a non-player, as he was North's runner in his distinctive fluro orange tracksuit top.
During the 60s and 70s Dwyer ran a sportgoods store in Pascoe Vale and entered into the hotel business in Brunswick in the 80s. Two of his sons also played for North Melbourne, Anthony and David. He won the Jack Titus Award in 1999.