The 20-year-old defender, on loan from Norwich, was reduced to tears in Friday night's 5-1 defeat at Anfield after something was apparently shouted at him from the Kop.
Merseyside Police and Liverpool, who have only just finished dealing with the fall-out from Luis Suarez's eight-match ban for racially abusing Manchester United's Patrice Evra, are continuing their investigations and Furman believes the perpetrators should be severely punished.
"I hope the authorities take heavy action because it was a disgrace," he said.
"He was abused and I'll leave it to Tom and the police to investigate that. What happened should not happen in the game."
Furman declined to confirm the insult was of a racist nature but after a police statement on Friday night clarified no-one had been arrested for racially aggravated behaviour his former team-mate Rueben Hazell claimed it was after speaking to Adeyemi on Friday night.
The Shrewsbury defender wrote on Twitter: "After the game I went into the Oldham dressing room to catch up with my old mates and spoke to Tom Adeyemi directly about what the altercation with the fan was all about.
"He said he was racially abused by the fan. This is FACT.
"He also said that police want to speak to him and was asking 4 advise (sic) from his team-mates what to do.
"Every1 around him told him 2 go ahead and give his version of events to the police.
"It's time for this kind of behaviour to stop. This isnt acceptable in football or in society and its time 4 black players 2 show a stand like young Tom did.
"As a 20yr old I thought he managed the situation extremely well and it was sad that the incident reduce him 2 tears in what should of been the happiest day of his footballing career to date."
Officials and players from both clubs would not comment on whether the abuse was of a racist nature but Furman hinted as much.
"It is difficult for me to say anything. It has been highlighted over the last few months and it is an absolute disgrace," he told BBC Radio Manchester.
"Tom, the authorities and police will sort out the situation but a lot of credit goes to Tom for the way he handled the situation."
Liverpool, Oldham and Merseyside Police are all working together in an attempt to discover the circumstances surrounding events last night.
On Friday night the club issued a statement which said: "An incident occurred in the second half of the match which is now being investigated by both the club and the police.
"We will continue to work with the police to establish the details of what actually happened and will make a further statement in due course."
Merseyside Police have little to add to their overnight position, when a spokesman said: "Merseyside Police can confirm that no-one has been arrested on suspicion of racially aggravated behaviour during the FA Cup fixture at Anfield.
"However, the Force and Liverpool Football Club are investigating following an incident that occurred during the second half of the game.
"The aim of the investigation is to establish the full circumstances surrounding the incident to ascertain if action needs to be taken."
It is understood police spoke to Adeyemi after the match and there talks between Oldham and Liverpool after the game.
The League One club will not comment further on the matter until they have received an update from police but they are happy with the way Liverpool dealt with the matter immediately and the importance they placed on the player's welfare.
Adeyemi's parent club Norwich spoke to him on Friday night to offer their support and have asked Oldham to keep them informed of developments, while it is understood they have also received assurances from Liverpool the matter will be investigated thoroughly.
Norwich chief executive David McNally wrote on Twitter: "Will pass on your support for Tom. He is ok.
"LFC have assured us that the awful incident will be fully investigated."
