The only way turning on the key should effect this is if the turn signal for either side is on. Power for the 4 way is battery power and should be present at the flasher all the time, key off or on, power goes from the flasher to the switch, trace that wire and conform. There is a function internal in the switch, with key on, that if 4 ways are on and you turn your switch to left or right signal, the 4 ways are by-passed and that signal should flash. I have found a couple times where the switch is not doing this correctly and is cancelling the 4 ways with key on even though neither turn signal is active.
With key on, turn signal lever in center of off position, turn on 4 ways and just wiggle turn lever, sometimes this will make the 4 ways work, if so the switch is not fully centered all the time not allowing them to work proper. if this is the case,loosen the 3 screw's holding the switch and wiggle it around, re-tighten and see if this fixed it, if not, live with it or replace switch again.