Running on empty there is some strange opinions with the answers so far ---for the last 20 years or so most fuel pumps are electric motors and in the tank ---with the fuel being the lubrication and cooling median--so running out means for a perioid of time there is no cooling and no lubrication---bad newes for an expensive part
The lower the fuel level is the more room water has to evaporate out of the gasoline. This causes rust in the fuel tank and when you do run out of fuel you're running on all the nasty stuff that's collected at the bottom of the fuel tank. My 78 mustang originally held 13 gallons. It now holds 11.5 because of all the sediment and rust that has formed over time. You don't want your engine running on all that gunk. Best way to prevent this is keep you gas tank as full as possible.Although I am an offender of this principle, here is how I understand it: Keeping fuel in tank will help keep the electric fuel pump cool as heat will harm electronic components. The bottom of the pump has a sock or a filter on it to prevent the pump from sucking up rust or other particles. The fuel is further filtered in most vehicles with an inline filter. It isn't as common of a problem as it was in the mid 80's to mid 90's due to a tighter sealed tanks or plastic tanks. One problem that happens with a tank that has a build up of "garbage" is a lack of power after running for a period at higher miliage. The sock becomes clogged and starves the engine of fuel. Usually when a tank has debris of rust and such, the car is usually in pretty rough shape.
I tend to not worry about it, running almost to empty (about 20 miles after the light comes on for mine). I avoid getting gas if the tanker is filling the station's tanks as this will mix up the fuel and increase the chance of getting contaminants. I also weigh in the fact that when you carry more fuel you carry more weight (approx. 7 lbs. per gallon), thus decreasing your fuel economy.
I try not to run below half because like one above says it keeps the pump cool and extends the life saving you $due to dirt etc that can gather in fuel tanks it is better not to run them to empty. unless is an old car, there will not be rust as cars generally have plastic tanks these days also remember, it costs the same to fill the top half of the tank as it does to fill the bottom half a full tank really works for you in the case of an emergency without you having to worry bout running out of fuel
Another thing to keep in mind is that when the tank is full the fuel pump doesn't have to work as hard to deliver the fuel to the engine. causing less wear on the pump itself.
Dirt is not the main reason on cars with in tank pumps. The pump is cooled by the fuel. Anything under 1/4 tank is possibly causing the pump to run hot. Doing this a few times is not major, but running your vehicle under 1/4 tank on a normal basis will cause the pump to fail. They usually fail without warning. The most common place of failure is the gas station because the cold gas shocks the hot pump.