A spokesman from the prosecutor's office in the southern city of Constance said it would investigate after a number of complaints were made, according to the German news agency DPA.
Under German law, publicly displaying the Nazi symbol is illegal.
Theatre-goers in the picturesque lakeside city have been offered free tickets to Friday's premiere if they wear a swastika, given to them on entry, during the performance.
Those who pay for a ticket will be asked to wear a Star of David “as a sign of solidarity with the victims of National Socialist (Nazi) barbarism,” the theatre's operators wrote on their website.
The premiere of Mein Kampf, a reference to the book written by Hitler when he was in prison before taking power, is scheduled to take place on Friday April 20th, the anniversary of his birth in 1889 in Austria.
The theatre says it aims to show how easily people are corruptible, but the approach has sparked controversy.
The German-Israeli Society in the Lake Constance region and the Judeo-Christian Society called for the boycott.
The play by George Tabori, who died in 2007, is a caricature of Hitler's youth, according to the theatre.
Investigators, however, need to examine the extent to which this can be a freedom of artistic creation.
The controversy comes as German authorities are worried about the rise of anti-Semitism, especially in schools.