The group released pictures of the execution on a square in Nahyat al-Alam, a town a few kilometres north of Tikrit.
One picture showed what the IS statement called "talismans" found in the victim's possession but appear to be nothing more than prayer beads and a green Shiite banner.
Some residents who spoke to AFP said the executed man was a Sunni who had recently joined the police in Samarra, a town further south which is under government control.
Signup to our newsletter and follow us on Facebook and Twitter!
But a Kirkuk-based cleric who knew the victim said he was a Sufi leader with a following in the Naqshbandiya brotherhood, whose main power base has always been in the Tikrit area.
The shadowy organisation, led by senior officials of the former Baath regime, joined forces with the jihadists when they swept across Iraq's Sunni heartland in June.
The Naqshbandiya's adherence to mystical Sufi practices clashes with the traditionalist interpretation of Islam IS claims to follow.
The executioner was identified by some local sources as a top IS operative for the central Salaheddin province, of which Samarra and Tikrit are both part.
On Monday, IS militants executed 13 men described as members of the "Knights of Al-Alam" -- an anti-jihadist Sunni tribal group -- on a roundabout between Tikrit and Nahyat al-Alam.