When apartheid was in full effect, South African Airways was forbidden from flying over most African countries.
As a result, SAA had to fly around the continent instead of over it, making most journeys far longer than their competitors. Flights to Lisbon took 2.5 hours longer than the normal route. Flights to Athens took 4 hours longer. Almost always a refuelling stop was necessary, adding even more time to the journey. Thankfully, the wretched days of apartheid are behind us.
Interestingly, TAP of Portugal had the same problems because they were prohibited from flying over most OAU (Organization of African Unity) countries. This was "due to the Portuguese government's refusal to grant their colonies independence." Once the colonies were granted independence, the overflight bans were lifted.
Indeed there are many examples of geopolitics causing airlines to make huge alterations to their flightpaths. There is a wonderful collection of old route maps illustrating these alterations here.