The poll by teaching union NASUWT suggests almost half (46%) of teachers have been asked to cover lessons for absent colleagues.
The union’s boss Dr Patrick Roach has warned higher rates of staff absence are making "a very challenging situation much worse" and that teacher shortages are likely to rise.
Pupils began returning to class last week after the Christmas break.
New advice for secondary school and college students in England to wear face coverings in classrooms has been introduced.
The poll of nearly 7,000 NASUWT members in England found 23% said staff absences due to Covid-19 were having a major impact on their school while 61% said they were having some impact.
Around one in eight (12%) of staff surveyed said guidance on pupils wearing face masks in classrooms was not being followed and 10% said there was no effective system in place for testing pupils onsite.
Dr Roach said: "It is very concerning that our members are telling us that staff absences due to Covid-19 are having serious impacts on teaching and learning.
"Higher rates of staff absence are making a very challenging situation much worse for schools struggling to maintain appropriate staffing levels without disrupting pupils' education.
"Whilst the start of term saw around one in 10 teachers absent due to coronavirus, these numbers are likely to increase in the absence of effective measures to ensure Covid-safety in classrooms."
Around two in five (44%) teachers said their school had a plan in place for deploying Co2 monitors while 18% said there was no such plan.
He added it was “disturbing” that some teachers reported their schools had no plans for Co2 monitors, saying “urgent” investment is needed to provide air filtration units in classrooms where needed.
"Inviting schools to bid for the limited number of air purifiers that are being made available by the Government is simply not good enough,” he said. “The safety of pupils and staff in classrooms should not be a lottery."
A DfE spokesman said: "Schools across the country reopened last week and staff are working tirelessly to ensure classrooms are safe for face-to-face learning, and despite the challenges in the first week of term, millions of pupils have returned to be with their friends and teachers.
"We've supported schools to continue classroom teaching for pupils through encouraging former teachers to step in, and extending the Covid workforce fund for schools that are facing the greatest staffing and funding pressures.
"We've also asked schools to have contingency plans to maximise attendance and minimise disruption to learning, should they have high rates of staff absence, and are working with the sector to share case studies of flexible learning models to support the development of those plans."
The department will release its pupil and staff attendance figures for the start of term on Tuesday.