Some diesel engines will have problems, others will not. It all depends on how the injection system works on the vehicle. The bigger issue to worry about is the extraneous parts, such as the fuel pump (primary or low-pressure and secondary or high-pressure). Diesel is primarily an oil, which helps cool and lubricate the parts it comes in contact with. Those parts include the pumps (as mentioned) and the injectors. When you run out of fuel, parts tend to heat up, more-so the pumps than the injectors, because at least the primary pump will continue to try and pump with having the aide of the fuel to cool it. This is actually true of most in-tank fuel pumps used in fuel injected vehicles whether it's diesel or gasoline.
Bottom line is, don't run your fuel low, as you can be causing your fuel system issues. It probably won't hurt the engine itself, though.
Older diesel systems had to siphon fuel from the tank. If they ran out of fuel, it was heck trying to get them started again. You had to bleed the air from the system so it would siphon again. You shouldn't have to worry about this with newer vehicles (whether gas or diesel). Like I stated earlier, they use one pump, usually located in the tank to provide the fuel to the secondary pump, where it is pressurized to a much higher level. A lot of diesel vehicles also use more than one fuel filter to ensure the fuel going into the injection system is pretty clean. The tolerances on the secondary injection pumps are pretty tight, so foreign matter in there could reek havoc.