A second equally important reason not to shut down an engine on fire during the initial take-off phase (apart from the risk of shutting down the wrong one) is that it is actually still producing thrust!
Following is more or less a brief overview of the concept we us at Sabena on al our planes from B737 to A340.
Before all take-offs (at the gate) the crew has to brief what to do in case of an engine faillure after V1, including at least:
-) who will be flying and who will be clearing.
-) what routing shall we follow (straight ahead if possible, or a curved escape route as published by our company) (BTW, SIDs are never flown in case of emergency due to their complexity; we try to keep it simple)
-) altitudes like Minimum Sector Altitude, Minimum Holding Altitide, Heighest Terrain etc.
-) are we good for an immediate return to an airport or is holding required due to overweight for landing?
-) if weather at our airport of departure is too bad for a landing, where do we go to?
An engine fire right after V1 on take-off means:
1) kill the bell,
2) continue the take-off and use the autopilot asap.
3) pilot flying flies the plane as briefed,
4) at for instance 1000ft, split the flightdeck in a pilot flying (PF) and a pilot clearing (PC),
5) the PC does the recall items from memory whereas the PF manages ATC, flightpath and a return to the airport for landing or a holding clearance in case of overweight.
6) when recall items are complete (and fire is out), the cockpit is back to a normal concept with the Pilot Non-Flying (the PC that was) completing the checklists and the PF giving a brief update on the position and configuration of the plane.