Will a faulty sending unit shut off the engine?

No, a faulty fuel sending unit will not directly shut off the engine. The fuel sending unit's primary job is to measure the level of fuel in the tank and send that information to the fuel gauge on the dashboard. It doesn't control the flow of fuel to the engine. A malfunctioning sending unit will result in an inaccurate fuel gauge reading (or no reading at all), but the engine will still run as long as there's fuel in the tank and the rest of the fuel system is working correctly.