YES

Definitely these trees can grow over forty feet.

... And the roots can spread at an alarming rate too. Their roots grow fast as they look for water. They will seek out cracks in rock, including concrete. Birch tree roots are aggressive and they look for water they will pry open cracks or joints in sewer pipes. Birch tree roots are among the most aggressive and destructive tree roots.

A rule of thumb in determining a trees root spread Billybob is that in order to support the tree, the tree roots will be at least as wide, if not more, as the width of the trees canopy.

When you say "four feet away from our property" do you mean four feet away from your boundary or from your house.
It could be 4ft from you boundary but 40ft from your house - in which case it isn't likely to cause much of a problem. But 4ft from a house wall isn't giving the tree enough space to grow. Even if you ignore the roots the canopy will want to spread.

I would suggest that you speak to your neighbour and agree a location for the tree while it is still a young sapling that would withstand moving - perhaps where both of you can admire it.

Benefits/problems of birch roots are variable - some are deep rooted and draw up valuable nutrients which get recycled to surface soil when leaves are shed. One potential problem is that some birches can be prone to aphids and so will spit honeydew everywhere.

Whether its 4 foot from your house or boundary it will still be a problem. As suggested, I would have a word with your neighbour and ask him to move it. If it is near your house it will cause problems with the foundations and if it is only 4 foot from the boundary it will, in time, overshadow your garden. If he doesn't see sense, then I would ask him to imagine what it will look like when you cut off all the overhanging branches and he ends up with a trunk and only half a tree.

Birches are quite shallow rooted so they will spread into your garden, Also they do get into drains - we didn't know this until it happened to us. So basically it depends how big your garden is and how far away from the house and services it is. But in the right place they are beautiful trees.