Mr Burnham said there was "absolutely no doubt" that the electronic devices were a "gateway product", and questioned whether it was acceptable for them to be exempted from the ban on cigarette advertising.
The shadow health secretary was speaking shortly before the publication of a Labour public health paper, which he said would set out an "unapologetic, firm and decisive" case for state intervention to improve children's diet and tackle problems with smoking and cheap alcohol.
Mr Burnham accused ministers of "a real loss of leadership on public health", including by failing to implement plain packaging for cigarettes.
"We're saying it's absolutely right for the state to intervene — and probably do so even more decisively than we did when we were in government — to protect children," he said. "Because children don't control the situations they're exposed to, the environment they're exposed to or the food that's put down to them."