Having flown different types, some requiring TOGA thrust after an engine 'failure', some not, I would say there are arguments in both directions.
In the lower powered aircraft, it seemed like a good idea - get away from the ground because you'll soon be on one less. In some of the later turbofan twins, you were never short of power and full thrust could actually make handling more difficult.
Scenario: Light A/C, full derate (flex). V1/Vr probably Vmcg/Vmca limited. Adding asymmetric full thrust will put you near the edge of the controllability envelope for what gain? If you are fully derated there is ample fat in the system. I appreciate that the original question referred to a 'fire' as opposed to loss of thrust but a very high rate of climb is not always the best thing to have, especially if there is a low level off. There are some nasty little traps set for you if you end up in one of the 'altitude capture' APFD submodes which manufacturers love to include on their latest and greatest. These can be coped with but remember you have probably only just put an A/P in and are now concentrating on shutting down the 'right' engine, initiating an emergency turn, etc. I believe an Airbus (A340?) was lost during testing in similar circumstances.
To summarise: If TOGA thrust is a SOP, then follow it. If it is discretionary then I would advise caution at low speeds/weights. If you are not happy with the aircraft performance then add power until you are...
Aside: Some years ago the company I work for upgraded to a turbofan variant of a particular A/C but carried over most of the SOPs from the older type, including mandatory TOGA power. About a year later I heard that that at very low weights the FMC V-speeds could end up close, possibly below Vmca (full thrust) due to the FMC recalculating a new Vmca based on the derate. I don't know the truth but it was an interesting proposition: lo and behold suddenly new SOPs came out with minimum V1s and the removal of mandatory TOGA application...