Can I legally unplug or remove anything my neighbour has plugged into my outdoor outlet without getting into trouble?

 For god’s sake, grow a spine! They’re stealing your property. Are you going to let them enter your house and eat your food next? So long as you’re sure that you’re the bill payer and it’s your property; it’s yours to do with as you wish regardless of the situation. Even if it’s a life-threatening situation, you still have the right to unplug and they should have probably told you beforehand. But it’s very rare that this is the case. You might want to see if your neighbour has any problems with their supply first just to be on the safe side. Otherwise, go ahead and remove the cord and place it back on their property. If they complain, ask why it’s there in the first place. Make it clear that this isn’t OK. If they’ve asked your permission, or you already have an agreement in place, you’ll need to let them know the reason that you’ve changed your mind first.