Do high foot arches cause any problems?

Yes. Although the definition is subjective, high arches (known clinically as "pes cavus") are associated with a variety of problems related to foot supination and relative inflexibility. Ankle sprains are most common, ankle impingement and stress fractures of the metatarsals and calcaneus (heel) are also more common than in "normal arched" individuals.
On occasion. A high foot arch is a more stable foot structure. This type of foot sructure will require more support such as an orthotic than the average shoe allows. Further it will decrease the shock absorbtion that your foot will naturally allow compared to a foot with a lower arch.