We should first define what we understand by school vouchers. AFAIK Tuneman is arguing for some kind of education stamps, that government would hand out to poor parents, and a selection of schools would be able to redeem them in the form of the tuition.
As I understand it, that’s not what is usually understand as school vouchers. Not in a small liberal group that I am part of, at least. They argue school vouchers as a different system to finance schools, in order to introduce choice. That is, instead of having some ammount of the tax revenue allocated to the Education Ministry, they’d instead be used to issue school vouchers. Generally, they would be redeemable for the same ammount, but some argue vouchers for less populated regions should have higher convertability. Anyway, these vouchers would be sent to parents, one per each kid of course, and they would then use them to pay the tuition for their kids in the school of their choice.
Depending on the exactly details (like if government would force a specific curriculum, or simply a set of goals, that would be enforced through national exams), the advantage of school vouchers is that private schools would be integrated into the system (which I think is more sensible than simply privatizing public schools overnight). By having things de-centralized, we could see schools targeted for very bright kids, for kids with learning disabilities, etc. Schools would have the possibility of hiring teachers by using different standards, and parents could simply walk away, if they dislike a teacher, and the school doesn’t do anything about it. (Parents beating teachers, out of frustrating, is a big problem here.)
So, I think how well it works out, very much depend on the associated regulation. For instance, in my view, if you let schools, that adhere to the voucher system, to have tuitions higher than the voucher, then a lot of schools will just use it as a subsidy. I’m afraid parents don’t account for vouchers when they are valueing a school, so they will distorce prices, as schools will just ask for the voucher plus a certain ammount that parents are satisfied to spent.