An honest but unreasonable belief may absolve someone from criminal liability when it directly negates the requisite mens rea for the crime. The belief should be sincerely held and relate directly to whether the defendant knew their actions were unlawful.
For example: If someone genuinely believes that they have the right to take property because they are unaware it belongs to another — perhaps due to incorrect public records — their unreasonable but honest belief could negate the mens rea for theft.