His warning comes after a union member won pound;50,000 compensation, plus legal costs, against an unnamed education authority for a psychiatric injury.
Mr Smith believes it is "a ground-breaking case".
Stress was difficult to prove but authorities would be facing expensive bills if they were found to have neglected their responsibilities. Teachers who went through the claim process would be unlikely to work again, he said.
Mr Smith said: "Occupational stress is a major problem facing teachers and lecturers. The EIS has always been aware that the legal hurdles on stress cases have been particularly difficult to overcome. This particular out-of-court settlement is a small, but significant, victory for the EIS."
Employers should be using the Health and Safety Executive's management standards to work with unions, authorities and schools to cut job stress, he added.
Mr Smith said authorities and schools should be able to recognise the signs of developing stress and draw up plans to tackle it, including sharing workloads, adjusting teaching commitments, and putting in extra support staff. More flexible working arrangements might occasionally be needed.
During the past 12 months, the union has won almost pound;250,000 for those members who have suffered injury while at work. The biggest risk to teachers and lecturers is from slips, trips or falls.