I know you are, John, but your responses indicate some confusion all the same.
A takeoff based on reduced thrust is predicated on keeping that reduced thrust setting; the performance, takeoff distance, and climb performance that are predicted as part of the pre-takeoff calculations are based on continuing the takeoff with reduced thrust.
Certainly if one finds one's self in a situation requiring greater thrust, it may be increased. However, one may also find one's self in a position of having insufficient control authority if one has used the performance numbers for the reduced thrust takeoff.
Remember that increasing thrust asymmetrically increases yawing moment. One needs a certain amount of rudder (and aileron and spoiler deflection in many cases) to counter this increase. At low speeds, pushing power on the good engine may counterproductive. The takeoff has been planned to safely continue, and meet all the applicable climb requirements, with the existing reduced thrust.
Whether one is on fire or not, or whether one has a fire indication, doesn't change the basic requirement to fly the airplane. Unless the procedure for your particular airplane calls for increasing thrust early in the takeoff, then one should not do so. Whether the airplane is on fire or not doesn't change that requirement: fly the airplane to a safe altitude, clean it up, and handle the abnormality or emergency.
If a fire warning occurs, silence the warning, continue, clean up, and execute the memory items or checklist items as appropriate. The rules don't change: Aviate (fly the airplane), Navigate (direct the airplane), and Communicate (talk about the airplane). Do that.
If you're on fire, remaining airborne for an extended period isn't in your best interest. Fly the airplane, first and foremost. If you've lost thrust and have an engine-out (and burning), then you've got the same problem initially as with a simple engine failure. You're going to fly the airplane to a safe altitude, begin speeding up and cleaning up, address the problem, fly the turn procedure or navigate otherwise as required, and notify ATC or other resources as necessary. If you're on fire, you'lll be likely returning to the airport very shortly. In order to do that, you need to fly the airplane.
Your question asks about an engine fire, and increasing thrust on the burning engine. Clearly, if one isn't going to increase thrust with an engine failure, then one isn't going to increase thrust with an engine fire.
Our procedure in the event of a power loss during takeoff, after V1, involves climbing to a pre-determined minimum level-off altitude, and stopping the climb. We begin cleaning up the airplane, and with the initial increment of flap retraction, we set max continuous thrust. We then clean up the airplane, address the problem, run the appropriate checklists, and in the meanwhile navigate as necessary while requesting help.
What we don't do is increase thrust. Reduced trust takeoffs are constructed with a power loss in mind. It's already taken into account insofar as runway takeoff distance required, and the climb performance once airborne. It's also predicated on other factors, each of which combine to either make it safe to do the takeoff with an engine failure at or after V1, or not. If not, then it's not a reduced thrust takeoff. If it is safe, then a thrust increase isn't required.
If you happen to fly an aircraft that uses a different protocol, then follow that.