Do LED and photodiode rise times matter for time-of-flight (TOF) measurement?
Generally the faster the rise time the more resolution you will be able to achieve as there will be less ambiguity on the receive edge.  It also does not make sense to use a detector that is significantly slower than the transmitter as this will waste the transmit rise time.  Note that the rise time is the important factor here, the pulse length does not really matter so long as it is long enough to reliably detect.  The length may be constrained by the duty cycle of the transmitter, however.  Some transmitters are pulse-mode devices which can emit a very powerful burst of light (as in a several watt laser) for a few ns, but then need to recover for several us before another pulse can be generated.  

It's not so much the rise time that you need to characterise; it's the delay.  The rise time just needs to be fast enough to get an accurate measurement, then you need to calibrate out all of the delays.