In her petition, Fure said the accumulation of plastic both detracted from the "serenity of the grounds" and created "a massive environmental issue".
It was discussed at a council meeting on 13 October, during which the council heard that the cemetery, which is a Grade I Site of Importance for Nature Conservation (SINC), was littered with plastic decorations.
Fure told the meeting that plastic items gradually broke down into microplastics, which stayed in the environment and polluted the neighbouring Hogsmill River.
A report by council officers responding to the petition said the authority's focus was on collaboration rather than formally enforcing the ban on plastics at the cemetery, according to the Local Democracy Reporting Service.
Its current regulations restrict plastic fencing and visitors are asked to remove cellophane wrappers from flowers.
The council said it was prioritising an educational campaign to encourage visitors to stick to the rules.
The report said staff were told to remove plastic and cellophane wrappers from flowers during routine maintenance and carry out litter picks in the cemetery twice a week.
Independent councillor James Giles, submitted a motion for the council to agree to a phased withdrawal of plastic decorations at the cemetery over three months.
It was rejected with 33 votes against, two in favour and three abstentions.
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