In some cases, yes. Although courts are generally reluctant to place the financial burden for private schooling on the non-custodial parent, it's not unheard of by any means. If your child has special medical or educational needs then it's more likely that a judge would allow the cost of private schooling to be added to the support obligation.
Most courts are also reluctant to force a non-custodial parent to bear the costs of a religiously-oriented school, possibly since this would amount to judicial (and by extension, State) "approval" of a particular religion. Again, this isn't unheard of.