Sports med running specialized foot and ankle physician here. "Arch height" is overly simplistic. There are flexible high arches, rigid high arches, flexible and rigid low arches, each with their own set of distinct "issues". To further cloud the subject, there are 3 planes of motion of the rearfoot and midfoot, which combine to create the motions of pronation and supination, which also govern arch and foot alignment. For example, there are people with high arches who overpronate tremendously, causing a twisted forefoot on rearfoot alignment, due to transverse plane dominant overpronation. The arches are high when these people are standing, but you'll see a huge bulge on the the instep. People with "flexible" high arches also often overpronate significantly. Their arches are high when seated but collapsed significantly from this alignment when standing. These feet will have issues that do not necessarily fall into the "high arch"/"low arch" black and white compartmentalization that is suggested herein. To further cloud the picture, as one poster suggested, is that feet run a spectrum of alignments and combinations thereof. It's just not that simple. Add leg, knee, and hip alignment, ligament and musculotendinous imbalances or contractures, running form/technique(ex., overstriding), body weight, running shoe choice, running surface, training errors, nutritional errors, underlying medical conditions (osteoporosis, etc.), training environment, lifestyle and occupational influences etc.,, and the influences on what kind of injury is related to "flat vs. high arch" becomes ridiculously murky.
BUT, all that being said, of the thousands of running injuries I've evaluated over the past 20 years, it has become clear that there are certain injury patterns more associated with rigid, underpronating high arched foot types and others more associated with flexible, overpronating foot types.
Rigid high arches: sesamoid injury, peroneal tendon injuries, navicular stress fractures, insertional Achilles tendinitis/Haglund's
Flexible overpronating feet: medial tibial stress syndrome (shin splints), medial tibial stress fractures, posterior tibial tendinitis, 2nd metatarsal stress fractures, plantar plate injuries of 2nd mtpj, 1st mtpj arthritis and stiffness.
All foot types: 3rd and 4th metatarsal stress fractures, Achilles tendinitis, plantar fasciitis, Morton's neuroma (often a shoe issue!), lateral ankle instability(maybe a little more with the rigid high arch foot type).
Feet with long 2nd or 3rd metatarsals/short 1st metatarsals (regardless of arch height): 2nd or 3rd mtpj plantar plate injuries, 2nd or 3rd metatarsal stress fractures.