When Max and Clara go to see Klimt's Beethoven Frieze, it is depicted as framed canvases mounted on wooden blocks, free-standing in the middle of the room. This is incorrect: in actuality Klimt painted the frieze directly onto the walls of the entrance hall of the exhibition. A frieze is by definition a painting, sculpture or other decorative element applied or attached to the walls of a room or building, typically just below the ceiling or cornice.
Also, the Beethoven Frieze would no longer have been on exhibit in 1906, when this story takes place. It was painted in 1902 specifically for the 14th Vienna Secessionist exhibition commemorating Beethoven that year, and was sold to a collector in 1903, at which point the sections of wall containing the frieze were removed from the gallery. The frieze was preserved but was not publicly exhibited again until 1986.
Also, the Beethoven Frieze would no longer have been on exhibit in 1906, when this story takes place. It was painted in 1902 specifically for the 14th Vienna Secessionist exhibition commemorating Beethoven that year, and was sold to a collector in 1903, at which point the sections of wall containing the frieze were removed from the gallery. The frieze was preserved but was not publicly exhibited again until 1986.
During the second visit to the morgue, the incision results in blood pouring down the drain despite the murder having occurred approximately 48 hours previously. Dead bodies don't bleed like this.
In the final scene in the cabin of the Vienna Ferris wheel Reinhardt is shooting Bruckmüller with a pistol. As the story takes place in 1906 this would have been impossible as pistols were first issued to the Vienna police in 1912. In 1906 Reinhardt would have been issued with a Gasser Police revolver in .380 (9mm).
In one scene when Reinhardt is walking down the street, modern cars can clearly be seen parked/driving past in the background.
The famous gilded bronze statue of Johann Strauss in the Vienna Stadtpark was not unveiled until 1921, fifteen years after the action is set.
Despite being set in 1906, the Vienna Riesenrad (constructed in 1897, and the name translates as "giant wheel") is shown to have only 15 gondolas when it would have had 30 gondolas. During WWII, the Riesenrad was damaged by allied bombing and was rebuilt after the war with only 15 gondolas to help extend its operational life (but with the gondolas having only even numbers which explains why there is a number 24).