Oak and mangrove fence gates are not flammable, but all the other woods are (except warped and crimson). This can easily be tested by putting all the fences and fence gates in a line, and setting them on fire and seeing that everything apart gets burned apart from the oak and mangrove fence gates.

This issue also affects BDS, and was found after dumping the attributes for each block through a mod and noticing minecraft:fence_gate and minecraft:mangrove_fence_gate have flame_encouragement and flammability values of 0.

Below are before and after pictures from setting the netherrack on fire.