does airplane mode stop phone from pinging towers? with recent riot's in Iran, there is saying that even with the phone in airplane-mode, it's pinging nearby towers, giving away location data and information such as that.
this is a really big security risk and may get people into jail.

is that true? if so, how would one fix this problem on the android device? does the phone being root or not make any difference?
Location is determined using one or more of the following


Mobile radio (mobile registering itself with nearby towers without mobile data being on)
Cellular data is on
GPS 
WiFi


For more details see this Android Tracking by vendor

Coming to your question if device is on Airplane mode (radio disabled) and GPS , WiFi are disabled as well (Note that WiFi can be enabled in Airplane mode ), there is no way your location can be determined

(However, there are reports as highlighted in this question that with GPS off also, location data is sent to Google, but that is a different question altogether - How do I stop my device from sending my location to Google every second, even when the location setting is off? )