The warning came as President Emmanuel Macron’s flagship measure became compulsory in all bars, restaurants, and cafés, as well as public transport.
“There will be strictly enforced checks, and in the first week they will focus on tourist and urban areas,” said an Interior Ministry spokesman.
“If there is doubt about identity, or the documentation being used, then the police will be called.”
Maximum fines for repeat offenders without an adequate pass can rise to the equivalent of £3,200, while six-month prison sentences are also an option for the criminal courts.
UK travellers are one of the biggest visitor groups to France – the most popular tourist destination in the world.
They have been told they can use NHS proof of double vaccination, or show they have had a negative Covid test within the last 72 hours.
But there was still widespread confusion on Monday, as police said they expected to be overrun by call outs.
While venue managers and transports staff are expected to check passes, they cannot carry out identity checks, and have to summon officers if there is a problem.
“We are here to enforce the law,” said Olivier Hourcau, of France’s Alliance police union. “We expect our colleagues to be mistreated or insulted by a minority who are anti-everything.”
The nationwide adoption of the passes coincided with the UK removing France from its amber-plus list, meaning those with two jabs no longer need to isolate on their return to Britain.
This led to a huge increase in the number of British arriving in France on Monday.
A French government spokesman denied reports the digital QR code provided by the NHS as proof of double vaccination had to be uploaded to France’s anti-Covid smartphone app.
“The British app is acceptable in itself,” he said. “If you don’t have a smartphone then you just need to print it out.”
There have been violent scenes in cities such as Paris over the past month, as some 200,000 people protested against the new passes.
However, polls show that the vast majority of French approve of them in a country where 66 per cent of the population now have at least one vaccination jab, and 55 per cent are fully vaccinated.