zinc - may damage the immune system
It also advised people against consuming more than 10mg per day of vitamin B6 unless they are acting on medical advice.
High intakes over a long period can lead to a loss of feeling in the arms and legs, the experts said.
In addition the group warned against taking more than 1000mg of vitamin C, 1500mg of calcium or 17mg of iron per day.
In high doses, these can all cause stomach pains and diarrhoea. However, the symptoms should disappear once people stop taking these supplements.
Thorough review
The FSA board is expected to back proposals urging manufacturers to reduce the dose of potentially harmful vitamins and minerals in some supplements or place warnings on packets at a meeting later on Thursday.
Sir John Krebs, chair of the FSA, said the report followed a thorough review of scientific studies.
"While in most cases you can get all the nutrients you need from a balanced diet, many people choose to take supplements.
"But taking some high dose supplements over a long period of time can be harmful.
"We are using an extremely thorough independent expert review of the scientific evidence on the safety of vitamins and minerals as the basis for new advice to help consumers make informed choices."
The supplements industry welcomed the report.
Dr Ann Walker, an advisor for the industry-funded Health Supplements Information Service, said: "In addition to encouraging a healthy diet in order to achieve good nutritional balance, supplements can play an important role in maintaining health where people are not getting all they need from food alone."