Have you (or anyone else) tested that? We know that wooden parts in pumps take heat damage just by being next to magma (without being submerged in it - if they are it's instant burn instead of slow damage over time), so I wouldn't say that wooden axles don't have the same problem without testing it (it'd be logical for them to take damage. Then again, DF isn't always the most logical). Non-magma-safe but fire-safe parts don't have that problem, so unless there's magma actually in the same tile, it would hold without problem.
I tried to perform an experiment on this, but I had a damnable time just trying to get wooden pumps to ignite. The conditions under which they will or won't burn aren't well-understood.
I'd like to see an experiment that runs a wooden pump right into the input of a magma-safe pump, so magma is never left next to the wood pump; if it burns, then wooden pumps are damaged by the act of pumping magma rather than the state of being adjacent to magma. You could also run magma up against the back of a dummy wooden pump that isn't actually being used for pumping. Closed wooden doors shouldn't have a problem, either. (Although I've seen unlocked stone doors appear to spontaneously open while exposed to magma - this also needs further investigation)
Anyway, the thing is, axles don't have a solid tile like screw pumps, so it's not possible to put an axle next to magma without putting it in a place where the magma can flow onto and submerge it.
I've allowed wooden axles to contact pumps that are actively engaged in magma pumping many times with no ill effects, so I'm inclined to call it safe until someone publishes an experiment showing otherwise.