1901 census - now called Harry and aged 24 so born 18
77 
I think he's already begun to take a few years off his age, and the marriage age of 29 in 1908 isn't so far out. Are you sure you have the right death cert, and if so, does it perhaps say aged 71 not 77?
1901 RG13 Piece 4676 Folio 59 Page 20
Pump Row, Framwellgate Moor, Durham
in household of David & Ellen Sykes
Harry Taylor boarder single 24 Coal Miner (Putter) New Durham, Durham
Going by the 1881 census with his parents giving his age as 9, which seems to be the most accurate, I'd be tempted to go for this birth cert, asking the register office to check his parents' names before you buy it
Jun 1872 Durham 10a 455 Taylor Henry
Kind regards,
Barbara