No, if the phone is in airplane mode it will stay that way when you turn it on.
But getting back to your problem, what data specifically is being sent when you have mobile data off. The only things that can be sent are calls and text messages (which do not use mobile data).
What might be causing your charges are the activation and control texts for iMessage, which go to a server in the UK (country code +44). These do not use mobile data. The way to fix that is to disable iMessage and FaceTime.
If your mobile network provider is not an Apple partner, that means they can charge for the international SMS messages that are used to activate and maintain your FaceTime and iMessage. Unless you want to pay for daily international text messages to Apple’s server in the UK you should log out of FaceTime and iMessage. Note that turning them off is not sufficient; you must actually log out.
FaceTime: Go to Settings/FaceTime, turn it on if it is off, tap on your Apple ID and tap Sign Out
iMessage: Go to Settings/Messages, turn it on if it is off, tap on Send & Receive, then your Apple ID, and tap Sign Out
Two more international texts will be sent after you do this, but they should then stop.