The amount of tax a British expat living in France has to pay will not change as a result of Brexit negotiations as tax agreements are negotiated with individual countries not with the EU as a bloc.
The double tax treaty, which ensures that people are not taxed twice on the same money, is a direct agreement between Britain and France. Post Brexit, however, British treatment of EU nationals living in the UK could effect the way British expats are treated in France and elsewhere in the EU. Countries such as Spain and France could introduce new tax rules specifically targeting British expats.
If Britain decides to remain part of the single market (the EEA, European Economic Area) nothing is likely to change as tax rules are the same for residents of the EU and the EEA. Inheritance tax, local property taxes and stamp duty on property purchases would also remain the same because equal tax rules for locals and other Europeans are part of the principle of the free flow of capital within the single market, enshrined in the 1957 Treaty of Rome.
However, in the event of a full Brexit – from the single market as well as the Union – it will no longer be necessary for EU countries to treat Britons the same as EU nationals and any advantages of being an EU national could be withdrawn. And, despite some cover being offered by the double taxation treaty, Britain would also be free to tax pensions that are transferred out of Britain.
It is also understood that chancellor (for now) George Osborne, with a possible recession on his hands, is interested in a UK version of the US FACTA (Foreign Account Tax Compliance Act) whereby British citizens would be liable to pay UK tax regardless of what country they live in.