Not unless the multipath causes the received digital pulses to fall outside what's called the guard band. In fact it can thrive on multipath. That's the principle behind DAB radio, there you have multiple transmitters all on the same frequency for a given mux. In principle that's no different to picking up a reflected signal along with the direct. Because the data stream is spread across multiple (thousands) of separate carriers, the data rate per individual carrier is very low, and the width of each pulse is quite long (in terms of time period). Therefore an additional pulse arriving on top of the 'wanted' (whether via reflection, or another transmitter) will have a zero or even constructive impact on reception, rather than destructive. That delay depends on a number of parameters that are applied, and the more data you wish to carry per mux, the shorter that 'window'.
For DAB, as long as another receivable transmitter is within a path distance difference of 80km, all is good. Clearly that's not really an issue with reflected signals, so you can imagine the immunity. For TV, the distance is shorter, about 20km, again unlikely to be a common issue. There is the use on a small scale of SFN (single frequency networks) for TV in some areas, and the distances can be adjusted by applying GPS defined offsets, to 'steer' likely interference zones out of the way. Also the COM 7 mux (and the recently defunct COM8) is a national SFN (all transmitters use UHF Ch 55). However, its payload has been reduced to allow for 80km working