Birch tend to produce shallow roots that spread out over a large area. The rule is that the extent of a tree's main root system extends out to what we call the dripline. If you dropped a line straight down from the most outer edge of the canopy, that is the drip line.
However, roots can extend further from a dripline, particularly if there is a concentrated water source. Roots will seek this source and concentrate their growth there.
If you're worried about the wall itself, that depends on the kind of foundation you have. The biggest problem is slab foundations that sit directly on the ground, and roots that grow big and fat beneath them can cause cracking. Slab homes often have their piping under the slab, and if there's seepage there, the roots can cause serious problems.
One of the best solutions is to go to the garden center and ask for root barrier sheeting. This is a geo-textile impregnated with tiny balls of root killer that causes root growth to stop cold when it comes in contact with them. Dig a narrow slot along the edge of your foundation where the tree is located and slide the sheeting down as far as you can. Then backfill the slot.