I received a speeding ticket at my home address in the UK via a car rental company in France. Does a UK resident have to pay for a speeding ticket picked up in France?

The owner of a car is responsible for the fines (parking, speeding, etc..) except when he provides the name of the driver - or of the person who rented the car. You can choose NOT to pay, and it is likely that there will be no further action. You will not hear of it unless you run across a French police check, in what case the police check into the national police computer will bring the case up.