Is a functional dependency on a "child" table allowed in third normal form (3NF)? Consider two simple tables, OrderHeader and OrderDetail which contain, respectively, the data shown below:

OrderHeader

| OrderHeaderID | CustomerID | OrderTotal |
|---------------|------------|------------|
| 1             | 124        | 14.5       | 
| 2             | 525        | 35.6       |
| etc           |            |            |


OrderDetail

| OrderDetailID | OrderHeaderID | ProductID |
|---------------|---------------|-----------|
| 1             | 1             | 415       |
| 2             | 1             | 52        |
| etc           |               |           |


The OrderTotal column can be calculated by adding the price of each product from the OrderDetail table.  That sounds like a functional dependency! ie:


SUM(OrderDetail.Product.Price) → OrderHeader.OrderTotal  


...but afaik it's not a violation of 3NF.  

Is this actually allowed?  If yes, is it disallowed in higher normal forms?
There is an informal dependency, but it's not a functional dependency.  See the wiki article for the formal definition. Here's an excerpt in simple terms:


  In simple words, if the values for the X attributes are known (say
  they are x), then the values for the Y attributes corresponding to x
  can be determined by looking them up in any tuple of R containing x.


There is no normal form definition which precludes summarizing child record values in the parent record.  However, lots of people would suggest against doing this in transactional databases, particularly where there's a possibility that a child record could be updated (or deleted, or more child records added) such that there's a risk of the attribute in the parent becoming inconsistent with the aggregate value of the children.

If your OrderHeader table had a CustomerName column, then that would violate 3NF because of the transitive functional dependency on CustomerID.

Being informal, a lot of people would say that OrderTotal is denormalized because it's susceptible to things that look like insert, update and delete anomalies and so people draw a comparison with the formal definitions of normalization and normal forms (since these are also about avoiding insert/update/delete anomalies).  There are some sticklers, though, who will be adamant that avoiding summary attributes in parent records is not the same thing as normalization per se.