"A preliminary investigation started yesterday," local prosecutor Patrick Quincy told AFP, as authorities opened a case file on a drunken night out by a group of British men in the French Alpine ski resort of Val Thorens.
The investigation was launched after a complaint from French anti-racism group SOS Racisme. Quincy said he had been unaware of the incident beforehand.
A lawyer for the restaurant where the party took place, La Fondue, said the establishment was also planning to file a complaint and that investigators had been to the restaurant on Thursday.
"We are preparing to hand in a complaint tomorrow," the lawyer, Julien Andrez, told AFP, adding that two charges were possible: "Inciting racial hatred" or "glorifying crimes against humanity".
Burley, the MP for Cannock Chase in Staffordshire, was sacked from his role as a parliamentary private secretary to Transport Secretary Justine Greening after pictures and video taken at the party earlier this month were published in the British press.
"They are launching a preliminary investigation and I understand I am not the focus of it," the 32-year-old told the BBC.
"I do not believe I have broken any French law and have distanced myself from the behavior of other people on the stage."
Burley was photographed sitting next to the groom, who was wearing the uniform of a World War II-era German SS officer.
'Offensive and foolish' behavior
A video showed a guest raising a toast to the Third Reich at the party and reports said the group had later chanted "Mein Fuehrer!" and the names of Nazis Heinrich Himmler and Adolf Eichmann, who were responsible for the Holocaust.
The Conservative Party said in a statement that Burley was removed from his post because of his "offensive and foolish" behavior at the party.
Cameron also asked for a full investigation.
In a message to his constituents on Wednesday, the MP voiced his "deepest regret", adding: "There are no excuses for my foolish behaviour which, over the past two weeks, has caused so much distress to so many people.
"I made a bad error of judgment and you deserved better from your local MP.
"Being involved in a stag party where an SS uniform was worn was wrong and offensive. It was the wrong decision on my part; crass and insensitive.
"I am deeply sorry, and want to take this opportunity to offer the people of Cannock Chase an unreserved, wholehearted and full apology for the terrible offence this incident has undoubtedly caused."
Burley said his family had "been through hell" as a result of the publicity surrounding the party.
And he insisted: "I have no sympathies whatsoever with Nazism, racism, or fascism... I personally did not participate in any alcohol-fuelled attempted toasts by other guests to the Third Reich.
"Nor did I participate in any chants, offensive or otherwise. As the video showed, I left the restaurant immediately when that inexcusable behaviour by other guests started."
Under French law it is a crime to make anti-Semitic statements or exhibit Nazi uniforms or emblems in public, unless required for a film, play or other cultural production.
Earlier this year, British fashion designer John Galliano was convicted of anti-Semitism for hurling abuse at bar patrons in Paris's Jewish quarter and given a suspended fine of €6,000 ($7,800).