I don't think so. The system controls only on the oil temperature. When the oil doesn't need cooling, there is no oil flow through the exchanger, and no fuel heating
On the Trent the FOHE is always on. There are no controls at all on the fuel flow side, and there is only a bypass on the oil flow side to enable to oil to bypass at very low temp (when the oil is below zero when starting)
The oil temp control system controls the AOHE which is normally closed, and opens at high oil temps to pass fan air through a heat exchanger to cool the oil.