Faith groups have raised fears that a new code of conduct could force teachers to promote homosexuality or other religions in contradiction of their own beliefs.
Concerns have also been voiced that a duty to "promote equality and diversity" could prevent church schools from recruiting leaders who belong to the faith.
The complaints have been raised in response to proposals made by the General Teaching Council for its new code, which will be binding on all teachers.
A consultation suggests that teachers, as well as promoting equality, should also challenge "discrimination, stereotyping and bullying, no matter the victim of or the perpetrator".
But in their responses to the GTC, both the Catholic Church and the Church of England said this could force teachers to act against their consciences.
Oona Stannard, the director of the Catholic Education Service, said that the call to challenge discrimination "could be used to oppose faith schools per se" by critics of religious schools.
Ms Stannard said her worries extended to the direction to promote equality.
"In religious terms, it would be unacceptable to expect anyone to be required to promote something contrary to their own faith beliefs and, indeed, it would not be possible for a person of faith to promote another faith - this is a matter of conscience," she said.
Instead, teachers should be expected to teach about other religions and to show them respect, Ms Stannard said.
The Church of England agreed that the new code risked creating a "conflict between teachers' professional responsibilities and their personal faith or philosophical commitments".
"As drafted, this potentially enables school leaders to expect teachers to act beyond the bounds of their conscience, particularly in relation to religious belief and behaviour or moral and ethical issues," the CofE response said.
The Christian Institute, an evangelical charity, has encouraged people to write to the GTC to complain about the proposed code.
Colin Hart, its director, said that many Christians were alarmed by the new diversity rules.
"The code rightly requires that teachers should respect their colleagues and those they teach," he said. "But respect for people as people is not the same as respecting or valuing every religious belief or sexual lifestyle.
"Forcing this on Christian teachers is to force them to go against their conscience."
The criticism of the code has been condemned by the National Secular Society, which has written to the GTC warning it that there is a campaign to "scupper" its "commitment to fairness".
Terry Sanderson, president of the society, said: "Who could possibly object to children being protected from the prejudices of bullies, bigots and zealots when they are captive in school?"
A GTC spokesperson said that the new code, due to be finalised later in the year, would require teachers to work within laws governing equality.
But she added: "The code does not oblige teachers to promote beliefs with which they disagree. On the contrary, it says quite explicitly that teachers must act respectfully towards children, parents, and colleagues in relation to their religion or beliefs."