Elliott, 43, has apologised after the photograph circulated on social media.
The Irish Horseracing Regulatory Board (IHRB) has started an investigation and the British Horseracing Authority (BHA) has banned him until it is completed.
"It is an interim decision which the BHA regards as proportionate in these circumstances," it said.
Elliott, who is licensed in Ireland, is one of the sport's most successful trainers after numerous triumphs at the Grand National and Cheltenham Festival.
Cheltenham, which is the pinnacle of jump racing and where Elliott finished as the leading trainer in 2017 and 2018, starts on 16 March.
Elliott has also won the Grand National three times, including twice with Tiger Roll. The Aintree festival takes place on 8-10 April.
Elliott said in a statement
that he took a phone call and sat down on the horse "without thinking", adding the image was taken "some time ago".
On Monday, he told the Racing Post
it was "indefensible" and that he was prepared for any punishment from the racing authorities.
"It was a moment of madness that I am going to have to spend the rest of my life paying for and that my staff are suffering for," he said.
"It absolutely breaks my heart to read and hear people say that I have no respect for my horses. That couldn't be further from the truth."
The BHA said it was "appalled" by the photograph and that Elliott's behaviour "deeply undermined" the industry's welfare values.
"We can't say we're looking after horses and giving them dignity when we're clearly not," eight-time champion jump jockey Peter Scudamore told BBC Sport's Laura Scott.
"There are people out there who don't like the sport and I can't argue with them when I see a picture like that."
Asked to recall his reaction to seeing the image, 62-year-old Scudamore added: "It just hit the bottom of my stomach.
"I think everybody in racing I know hoped it was fake, and then there was a slow realisation that it's not a fake. It's desperate sadness on so many fronts.
"It is just such an appalling image and I'm very sad about it. It was an act of crass stupidity."
Scudamore said he did not want to speculate on what he thought would be suitable punishment for Elliott, although there have been calls from some for the authorities to ban the Irishman.
The BHA's interim move has put Elliott's participation at the upcoming Cheltenham and Grand National meetings in doubt.
Owners of horses currently trained by Elliott are allowed to transfer them to a different trainer and run them at a British meeting "providing they comply with the relevant rules".
"I can't see a round of applause if his horse comes into a winners' enclosure," added Scudamore, who is now the assistant to leading trainer Lucinda Russell.
"Let's have an act of contrition, a massive donation to a horse charity. I don't know. I hardly know Gordon Elliott, but I don't think revenge is the answer.
"I just want some good to come out of it, give the dignity back to these creatures."
Elliott, who is based in County Meath, has apologised "profoundly" for the photograph.
"I can categorically state that the welfare of each and every horse under my care is paramount and has been central to the success that we have enjoyed," he said.
"The photo in question was taken some time ago and occurred after a horse had died of an apparent heart attack on the gallops.
"At what was a sad time, which it is when any horse under my care passes away, my initial reaction was to get the body removed from where it was positioned.
"I was standing over the horse waiting to help with the removal of the body, in the course of which, to my memory I received a call and, without thinking, I sat down to take it. Hearing a shout from one of my team, I gestured to wait until I was finished.
"Such background information may seem trivial at this time and will not allay the concerns of many people both within and outside the world of horse racing."
Michael O'Leary has said Gigginstown, who own Tiger Roll and a number of other horses trained by Elliott, will "continue to support him and his team, as they work to recover from this deeply regrettable incident".
He says the photo is "unacceptable" and "grievous" but calls it a "momentary lapse of judgement", adding "we all make mistakes" and that Gigginstown accepts Elliott's apology.
Cheveley Park Stud, who own several horses trained by Elliott including the unbeaten Envoi Allen, say they are "truly horrified" by the photo but will not comment further until the investigation by the IHRB is over.
The Jockey Club, which owns Cheltenham and Aintree racecourses, said: "Clearly this is totally unacceptable and not reflective of the respect and care that racehorses receive from participants in our sport.
"The anger and upset across racing says it all. We understand the authorities are reviewing this as a matter of urgency."
Betfair said on Monday it had chosen to end its relationship with Elliott, who had been an ambassador for the betting company.
"While we recognise that Gordon deeply regrets and apologised unreservedly for his poor judgement, his actions are completely at odds with the values of the Betfair brand and that of our employees," a Betfair spokesperson told the BBC. "With that in mind, we have decided to discontinue our association with Gordon with immediate effect."
The charity World Horse Welfare said: "This photo looks abhorrent. We understand the trainer has apologised and there is an investigation ongoing."
Great British Racing, the promotional body of British racing, said: "Respect for our horses is at the heart of everything that we stand for in British racing and the shocking image is counter to that and betrays the work of thousands of people loving and caring for our horses on a daily basis - we condemn it in the strongest possible terms."
An IHRB spokesman said on Sunday: "The investigation is ongoing and will be dealt with as quickly as possible."
Horse Racing Ireland also condemned the picture and supported the IHRB investigation, adding: "From a disciplinary perspective, the matter is in process, so any further comment on the matter or the detail of the case at this time would not be appropriate."
In the end, a vision without the ability to execute it is probably a hallucination.