Such values are ill-defined and vulnerable to misinterpretation, argued Southend head teacher, Robin Bevan.
The Association of Teachers and Lecturers (ATL) voted to monitor the policing of the requirement, introduced in England by the government last year.
Mr Bevan said he had concerns about how the values "might be interpreted by a future right-wing government".
"When it comes to the new requirement of promoting fundamental British values, including the role of law, here is one law that I would actively encourage you to disengage from", Mr Bevan, head of Southend High School for Boys, urged the union's annual conference in Liverpool.
The government brought in the requirement in the wake of the Trojan Horse allegations, which suggested there had been attempts by groups of hardline Muslims to take over schools in Birmingham.
Earlier this month a committee of MPs said that apart from one incident in one school "no evidence of extremism or radicalisation was found by any of the inquiries in any of the schools involved".
Requiring schools to promote "fundamental British values" could have "unintended consequences", according to the text of the motion debated by the conference in Liverpool.
"There is no one in this hall who would argue against the important role that schools and colleges play in promoting personal morality, in developing a sense of civic duty, in fostering engagement with our democratic structures or in embracing a wider global understanding," said Mr Bevan.
He said schools and colleges had, for a long time, been required to deliver a broad and balanced curriculum addressing issues of behaviour and conduct and placing this within a developmental framework of spiritual, moral and cultural understanding.
"These provisions have existed without controversy for many years and are ample and effective."
He accused the rules on British values as "deeply ill-considered" and "political posturing".
"In what way do fundamental British values differ, for example, from French or Swedish ones?"
They change over time, with women now "allowed" the vote, said Mr Bevan.
"If these fundamental British values change over time, we can hardly describe them as fundamental."
But, he argued, the problem was not just one of definition.
"Just take one moment to imagine how fundamental British values might be interpreted by a future right-wing government, or a partner in that government."
He said he was particularly concerned that Ofsted was being asked to gauge the views of students in order to assess how how well schools actively promote these values.
There had already been cases of "less-skilled inspectors" doing this in a "wholly-inappropriate way", he told journalists later.
In particular he was concerned about the difference between "what is taught and what is learned", with a proportion of every class apt to misunderstand or ignore lessons, be they about trigonometry or British values.
"I am not sure the government should ever be in the business of dictating values that should be taught in schools - but students should engage in those debates... plurality, that's the way it should be," said Mr Bevan.
Teachers should ignore rules on promoting "fundamental British values", a teachers' union conference has heard.