Running out of fuel is certainly bad for your fuel pump and can lead to the fuel pump being damaged.
When your gas tank gets low on gasoline the fuel pump that is submerged in the gasoline in the gas tank doesn't stay cool enough and then can overheat and then lead to failure of the fuel pump.
Even just running a bit low or putting a few gallons of gasoline in the gas tank can lead to damage of the fuel pump because the fuel pump requires the gasoline in the tank to keep it cool.
Low gasoline level in your tank also allows the fuel pump to suck in air which can cause damage to the fuel pump.
Fuel pumps in tanks are designed to last the life of the vehicle but can sometimes only last between 100,000 to 200,000 miles if you don't keep enough fuel in the fuel tank.
Overtime the fuel pumps do eventually wear out so if you have a vehicle older than 20 years and the fuel pump has never been replaced you really should replace the fuel pump to prevent failure of the fuel pump on the highway where it will leave you stranded and cause you to have to call a tow truck which will cost you the same or more than just replacing the fuel pump.
One of the best things to do to preserve the fuel pump lifespan is to keep the fuel tank on your vehicle at least 1/2 full or if you can keep the gas tank full completely to keep the fuel pump submerged in gasoline and prevent the fuel pump from getting too hot and overheating.