I'm more than a little surprised that searching for "fuel tank" and "heat shield" brings up no questions on mechanics.SE. Indeed the very keyword "heat shield" has not yet been added. (I cannot add it from my side.) In any case...
A car will typically have several heat shields. The ones along the center of a car are there to keep the feet of the occupants cool in the summer, and perhaps to increase the sound insulation.
The heat shield right under the spare tire (or "doughnut") keeps the rubber in that tire from melting, or disintegrating too soon.
The one I'm concerned about is the one under the fuel tank. When I look at how it is sculpted, I feel that keeping the fuel tank cool is only one of its design parameters. It also has channels that seem to lead any leaking fuel away from the exhaust pipe, with the drainage holes quite far from the exhaust pipe.
The idea then, presumably, is that even a drop of fuel (from a crack, from a tiny hole, from rust, or from stress fracture during a minor collision) will instantly ignite if it touches the exhaust pipe, possibly leading to a catastrophe (such as starting to bake the occupants inside or, worse, the tank exploding). A fire there seems risky because the driver will not even know, unless good samaritans happen to be nearby and honk like crazy.
It is not unusual to hear folks who discarded their rusted fuel tank heat shield say that they have been "driving for years without an incident".
As you see, I am already convinced that it's more than a little unwise to drive without the fuel tank heat shield. Yet it's quite common to hear people advising to drive without one. If you are in that camp, could you argue for that point of view? (You are hereby held harmless from any liability!) Even if you wouldn't do it yourself, feel free to act as a devil's advocate and illustrate how that argument might go.