The Pendle resident became an associate director in 2011 and helped the club's process of buying back both their stadium Turf Moor and the Clarets' training ground.
"I am proud to join the board of the club I have supported all my life and an institution that I have already demonstrated my commitment to over the years," said Nelson.
"The board has done a great job in positioning the club for a sustainable future and joining the team gives me an opportunity to support the club in a very practical way."
Nelson becomes the fifth member on the Burnley board of directors, and co-chairman John Banaszkiewicz and Mike Garlick added: "We are delighted to welcome Brian to the board of directors.
"This appointment represents the next logical step for Brian, given his close association with the club over many years.
"He brings with him not just a wealth of business acumen and contacts, but also a passion for the club and the town - a vital combination in helping the board deliver our ambitious strategy."