The findings of the survey of more than two thousand people, also shows 86 per cent were concerned terminally ill people would be offered assisted dying to save cash.
The research carried out by Christian charity Care was released exclusively to Premier on the day MPs debate the Assisted Dying Bill.
You can watch the debate live here.
It would allow doctors to prescribe a lethal dose to terminally ill patients judged to have six months or less to live and who request it.
It also requires the approval of two doctors and a judge.
Currently, assisted suicide is illegal under the Suicide Act 1961 and is punishable by up to 14 years in prison.
More than three quarters of the 2,082 respondents, 76 per cent, think terminally ill patients who feel they are a burden, would feel a sense of responsibility to access assisted suicide if the Bill is passed.
Some 63 per cent think 'some' patients in this situation would feel a responsibility to access assisted suicide, while 13 per cent think 'many' patients would feel this obligation.
The polling shows 16 per cent think no-one who is terminally ill and feels a burden would feel any sense of responsibility to use assisted suicide.
Supporters of the bill argue there would be enough safeguards to prevent abuse of the law but the polling data shows the majority of the public does not agree.
Some 23 per cent of people thought there was a significant risk NHS managers or politicians would prioritise assisted dying over end of life care to save money.
More than a quarter, 26 per cent, of people thought there was a moderate risk of this happening with 37 per cent agreeing there was a slight risk.
Just 15 per cent said there was no risk of this happening.
Care CEO Nola Leach said: "These results are a clear and stark reminder that no safeguard can protect against subtle and covert pressure and when you think that in countries and states where assisted suicide is already legal, the numbers go up and up, this really is a clear warning to MPs not to go down this route of doctor-assisted suicide.
"While assisted suicide is a deeply emotive issue, lawmakers have a duty to objectively scrutinise the evidence and it has often be said, hard cases make bad laws.
"Facilitating a situation where anyone feels they must choose assisted suicide rather than hospice care is incompatible with a compassionate and caring society."
She rejected suggestions the questions in the poll could have been staged to provide the desired response: "I would say they weren't angled in a certain way, they were very measured in how they looked at what people were thinking.
"The evidence is all there that people are concerned about this."
The Bishop of Buckingham, Rt Revd Alan Wilson, who supports the Assisted Dying Bill, said it would "give greater say to patients themselves in the way that their dying is handled".
"It's very important to have that protection for dying people in our law, and not just to leave things in the rather unsatisfactory discretionary way that they are right now," he told Premier.
Bishop Alan mentioned a case of a terminally ill hospice patient, who he said suffered unnecessarily because of the existing law: "The doctor that was available had simply decided that, for reasons of preventative medicine, he was unwilling to prescribe sufficient painkillers to relieve the pain in a dying patient. This strung things out by several days, the care team were very, very upset.
"There is a certain point at which you will end up giving somebody painkilling relief that may indeed shorten their life. Now all this Bill does is introduce the patient into the equation so that they ultimately have another option that right now they don't."
But leading palliative care experts have disputed the claim that doctors give doses of painkillers to shorten a life.
