The video, which is understood to have been filmed on a supporters bus on October 30, is a ten-second long clip and makes reference to Burns, who played for the club before going onto manage them, and Brown's sister.
The Scotsman has taken the decision not to share the video due to its offensive content.
Burns. who played more than 500 games for the Hoops, died in 2008 after a battle with two-year battle with skin cancer. Brown's sister passed away from the same disease just two weeks later.
A number of Rangers fans took to social media to call out their fellow supporters, with one writing: "They don’t represent me or my club, disgusting behaviour, which all true Rangers supporters condemn."
Another added: "I'm mortified seeing that, utter clowns that have no place in society let alone football."
A Gers fan on Twitter branded the video "diosgusting", adding: "I'm a Rangers fan and it's that type of song that I personally think is the most offensive.
"Much worse than any chant about religion or politics. Glorifying in personal tragedy is sick. Songs about dead fans, sick players or relatives is subhuman."
Jonathan Burns, son of Tommy, shared the video and wrote: "As I’ve said before, these people are not real Rangers fans and do not reflect real football fans, a majority of whom are good.
"These people could live to be 1000 and not achieve what [my father] or [Brown's sister] have."
Brown was hit with similar abuse in the wake of the last Old Firm match, when a fan waiting outside Ibrox asked him: "How's your sister?"
Rangers later issued the supporter with a life ban from Ibrox.