Overriding public virtual functions with private functions in C++
You do get the surprising result that if you have a child, you can't call foo, but you can cast it to a base and then call foo.
child *c = new child();
c->foo; // compile error (can't access private member)
static_cast<base *>(c)->foo(); // this is fine, but still calls the implementation in child
I suppose you might be able to contrive an example where you don't want a function exposed, except when you are treating it as an instance of the base class. But the very fact that that situation pops up would suggest a bad OO design somewhere along the line that should probably be refactored.
The problem is that the Base class methods are its way of declaring its interface. It is, in essence saying, "These are the things you can do to objects of this class."
When in a Derived class you make something the Base had declared as public private, you are taking something away. Now, even though a Derived object "is-a" Base object, something that you should be able to do to a Base class object you cannot do to a Derived class object, breaking the Liskov Substitution Prinicple
Will this cause a "technical" problem in your program? Maybe not. But it will probably mean object of your classes won't behave in a way your users expect them to behave.
If you find yourself in the situation where this is what you want (except in the case of a deprecated method referred to in another answer), chances are you have an inheritance model where inheritance isn't really modeling "is-a," (e.g. Scott Myers's example Square inheriting from Rectangle, but you can't change a Square's width independent of its height like you can for a rectangle) and you may need to reconsider your class relationships.