I am looking to fly mid-July on Qatar Airways round trip between Cape Town, South Africa and Paris, France. With the political situation at the moment, would there be any recourse for potential flight cancellations?
You'll be fine. Qatar Airways is flying more or less normally, albeit with slightly longer flight times than usual, and your flights are not to/from/via/over any of the countries imposing sanctions.
The only theoretical risk is that the current conflict deteriorates into outright war, but that's highly unlikely.