The lecturers' union NATFHE believes that Carol Hughes, 42, should not be allowed to study at the college because of her extremist views. She is currently on a management studies course.
Ms Hughes was one of three BNP councillors elected last May in the town where there is a large ethnic-minority community. Burnley was the scene of riots last summer and has proved a fertile recruiting ground for the right-wing nationalist party.
Colin Gledhill, Natfhe's North-west regional official, said: "We recognise everyone has a right to education and we agonised over the moral question of calling for her place at the college to be denied her. However, we think we are morally correct to contest that right for a person who utters views which can only be seen as inflammatory in a multicultural society."
He said the 160 members in the college's Natfhe branch had decided against strike action, but would do whatever was necessary to back a colleague who refused to teach Ms Hughes. There was also disappointment that the college had not acted.
Mr Gledhill said: "We believe that if the college mission statement to promote social cohesion is worth anything, then the principal John Smith should not tolerate elements in the academic community whose aims are completely opposed to that."
Burnley College said no one was available to comment on the issue.
A spokesman for the BNP in Burnley accused left-wing elements within the college teaching staff of trying to stir up opposition to Ms Hughes. "People of all differing viewpoints have a right to education," he said.
In 1984, Patrick Harrington, a National Front student organiser, needed a police escort to attend lectures on philosophy at the then Polytechnic of North London. Hundreds of students mounted pickets to stop him entering the building.