The tank, a decommissioned T-55 purchased from an army depot and painted with Red Star's logo, was installed with a crane outside the stadium stands as an "attraction", the Serbian club said on their Facebook page.
However, the move divided fans and drew strong condemnation in neighbouring Croatia, who saw the tank as a symbol of the assault by Serb forces on the city of Vukovar at the start of Croatia's independence war in the 1990s.
An article in the leading Croatian sports daily Sportske novosti slammed the "morbid provocation from Belgrade".
"They want to play in the Champions League as they celebrate one of the worst crimes," the newspaper wrote.
Online Croatian news site Index.hr accused Red Star of "displaying a Vukovar tank in front of their stadium", although neither the club nor its supporters presented it that way.
On their own fan website, redstarbelgrade.rs, supporters said the tank was meant to represent the "machine" of the club and the nickname of a group of hardcore fans, who are known as the Northern Army.
The controversy comes amid already high tensions after a Serb-owned bar in Croatia's Knin region was attacked by masked men while patrons were watching the first leg between Red Star and Young Boys last week.
The relationship between Croats and the Serb minority in Knin is still tense almost 25 years after the end of the 1991-1995 independence war that separated Croatia from Serbia and the rest of the former Yugoslavia.
The incident involving the tank divided opinion in Serbia, with some welcoming the move on social media while others denounced it as in "bad taste".
On Tuesday fans at the stadium clambered on top of the tank and posed for photos.
Questioned at a press conference, Serbia's Interior Minister Nebojsa Stefanovic said "there were no elements of a criminal act and therefore no reason for police intervention" in regards to the tank.
Red Star are hoping to secure progress to the Champions League group stage for the second season running after a 2-2 draw in the first leg in Switzerland last week.