The think tank's pre-election report, based on parties' spending pledges, says schools face up to 12% in real-terms cuts over the next parliament.
It says increasing pupil numbers and staff wages are adding to the pressure.
The Association of School and College Leaders said it presented a "bleak picture for education funding".
The IFS says day-to-day spending on schools has been relatively well-protected by the coalition government compared with other public service areas, but after the election that will change whoever is in government.
Analysing the spending pledges of the Conservatives, Labour and Liberal Democrats, the IFS said they all suggest a real-terms reduction of 7% per child by 2020.
When the extra costs of teachers' pensions, national insurance contributions and wage increases are included, the real-terms reduction in spending could be closer to 12%, the report said.
The Conservatives have said they will protect school spending per pupil, while Labour and the Liberal Democrats have committed to protecting the education budget for three to 19-year-olds - covering early years, schools and 16-19 education - in real terms.
But the IFS said: "Pupil numbers are expected to grow by 7% between January 2016 and January 2020, whilst economy-wide inflation between 2015-16 and 2019-20 is currently forecast to be 7.7%.
"As a result, the overall level of school spending could grow by similar amounts under the different proposals. However, if only just met, all these proposals imply real-terms cuts in spending per pupil."
In response to the report, the Conservatives said they were committed to delivering a good deal for schools, and would spend at least half a billion pounds more than Labour.
Shadow education secretary Tristram Hunt said his party had a better plan than what he called the Tories' "big cuts", while the Liberal Democrats insisted they would protect education budgets in real terms.
ASCL deputy general secretary Malcolm Trobe said:  "It is imperative that the government elected in May addresses this issue quickly and ensures that education funding is sufficient, sustainable and equitable.
"We recognise that there is considerable pressure on the national budget but the country must invest in education both for its long-term prosperity and, most importantly, the future of our children."
Russell Hobby, general secretary of  the National Association of Head Teachers, said that the golden age of education funding was coming to an end right at the time when the demands and expectations of schools were accelerating.
"These budget cuts come at a time when cost pressures on schools from pensions and national insurance are also increasing.
"Salary increases are not funded by the government and you can't keep reducing salaries in real terms and still attract talented people into the profession."
Schools in England will have less to spend per pupil over the next five years, no matter who wins the election, the Institute for Fiscal Studies says.