There is no "fundamental" reason why bump starting a car will damage any component any more than the usual starter motor starting, but that is with the caveat that the battery actually has enough power to power the engine and fuel systems as normal. And of course, the reason you usually bump start a car is because the battery is flat! If there is insufficent voltage to correctly operate the fuel system and EMS (usually requires >8volts) then the manufacturer cannot warrant what could happen, with things like partial fuel pressure, or stuck on injectors etc, all of which "might" cause damage to your engine.
So to play safe, they recommend you do not bump start the car, but jump start it (which obviously will ensure there is enough voltage for the systems to function). If you had say just a failed starter motor but the battery and rest of the electronics was fine then you could happily bump start your car no issues.
In fact, every time you drive your car you actually "bump" start it many many times, as lifting fully off the accelerator these days results in a zero fuel quantity injected, and the engine is actually not running (turning yes, running no)...... ;-)
Of course, if you have an Automatic transmission, you cannot bump start it, because it relies on hydraulic pressure to engage and hold a gear, and the hydraulic pump is driven by the input shaft, which is not turning (hence no pressure availible) with the engine off.
Manual gearboxes are not damaged by bump starting. Those fitted with oil pumps are really there for cooling purposes and lubrucation under large torque loads, neither of which is an issue for the couple of secs and low load of a bump start (an engine takes about 2kW to start it, my car puts over 150kW through the transmission most of the time without any issues!)
Finaly, DMF's, yes you could theoretically damage your DMF if you continuously bump started it for years on end, but it would be a slow wearing rather than a sudden failure, which would also probably be true to the clutch center plate damper springs.
Editted to add " old cars were of course easy to bump start, as the carb had a nice bowl full of fuel, so you didn't care if the fuel pump worked at key on, and the only electrics needed was about 2 amps to get the coil to fire the sparks, so even with a fairly flat battery that couldn't manage to drive the starter motor, you could get a decent bump start out of it. Modern cars now pull over 30 Amps as soon as you unlock the door (sometimes 100's of amps, no names mentioned LandRover, oopps, sorry.... ;-), so unless your battery is charged, the voltage will sag immediately, and no amount of bump starting will work"