A non-ideal battery is connected to a resistor. Is work done by the battery equal to the thermal energy developed in the resistor? Does your answer change if the battery is ideal?
A non-ideal battery has internal energy and hence the work done is equal to the thermal energy developed due to both resistances in the circuit and the internal resistance of the battery.
If the battery is ideal, then there will be no internal resistance and the work done will solely be the thermal energy developed in the resistor.