Yes, No or Maybe – come March 2019 (if it happens) you’ll cease to be an EU national, we may have partner agreement with France, or the EU, both or neither.
You may apply for a French passport now and then it wouldn’t be a problem, I’m told it’s fairly easy at the moment, if you decide to retain your UK nationality you’ll be able to come and go as you please… well, after queuing for several hours no doubt, Passport Control didn’t seem to be able to cope with ‘Summer’ let alone with suddenly have to check the details of EU Nationals.
If we leave without a ‘deal’ you may have to leave France immediately and hope the UK still accepts your ‘old’ EU passport to let you back in or you’ll be stuck in flux like Tom Hanks in ‘The Terminal’ it seems far fetched, but it’s a possible outcome, especially considering the Leavers and their hysterical fears around borders and passports, of course they’ll assume that all the new rules will only apply to Johnny Foreigner, but they won’t. Come Summer 2019 they might find their holiday plans thwarted because the EU doesn’t yet recognise the ‘New’ UK and our passports, so no Tourist Visas, that’ll be fun for everyone.