There are actually two separate myths surrounding bird feeder perches. The first is that a bird will get stuck to a perch if it’s freezing outside and the second is hummingbirds die at feeders because their body doesn’t produce enough warmth while sitting to drink cold nectar in the morning.
The first myth is absolutely false and the second one is most likely false.
A bird’s foot (and body) is well-adjusted to cold temperatures, so it seems pretty odd to think a cold metal perch would cause a bird to get stuck. The reality is that moisture is what causes something like your tongue to get stuck to a pole, but a bird’s foot does not have any sweat glands to provide that moisture.
Similarly, their feet and legs are made up of scaly tissue, so there wouldn’t be damage caused by cold.
The second myth arose after a few people found dead hummingbirds by or on a feeder’s perch. There are many reasons why a hummingbird could die at a feeder, whether being it’s time to go (hummingbirds don’t live very long) or having been stung by a bee. Hummingbirds often stop flying to incubate their eggs and sleep for the night when temperatures drop the lowest, so the idea that drinking cold nectar when stationary would kill them it pretty crazy.
However, since there is no scientific evidence that it doesn’t cause them to die (nor is there any that it causes them to), we can’t definitively say it’s a myth.