Under which circumstances would a politician be most likely to ignore the results of a public opinion poll on a particular subject when making policy regarding that subject?

I would say, if the poll is likely not to be reliable, especially if it is biased, then the politicians can safely ignore them. A very simple example is a poll sponsored by a group with a known interest in the question (e.g. imagine Lufthansa sponsoring the poll on "should the U.S. Government subsidize Lufthansa's flights in the U.S;))

Sheet stain on pee wee's sheets