Do I need a visa to enter Thailand multiple times as a British citizen living in France?

The airline you use from France may want to see confirmed flight out of Thailand within 30 days (as that is how long you’ll be stamped in for on arrival) if you have flights booked to Vietnam, or any other country, within that time frame, just show them when checking in in Paris.

Nope

No, with a British passport you get 30 days visa exempt each entry but you can only do 2 visa exempt entries by land border per calendar year. Are you flying when you exit as your airline may with to see proof of onward travel within 30 days to allow you to board for Thailand?

NO you don’t need a visa. You will get a 30 days visa exempt every time you enter Thailand. Just plan your Vietnam/Cambodia trips in between, so that no stay in Thailand exceeds 30 days.

No need for a Thai Visa.

You will Enter Thailand as Visa Exempt Entry and will be stamped in for 30 days.

That can be extended for 30 days More at local Thai Immigration office for 1900 Thai baht.

If you LEAVE THAILAND as you say, each time you RE-ENTER THAILAND you will again be STAMPED in for 30 days as Visa Exempt Entry.

No need to get a Re-entry permit as you will be stamped in VISA EXEMPT each time you arrive into Thailand.

If you are asked by the Thai Immigration Officers upon any arrival, just show the PROOF of your flight departing Thailand at the end of your planned stay.

However your airline at check-in may want to see a ticket departing Thailand within the 30 days you are initially STAMPED in on arrival.

Be sure to check the entry requirements for the other countries that you plan to visit.