Is snow piled against the outer wall likely to cause damp?
Here in Alberta this turns out to be a non-problem.  What tends to happen is that the snow slides off the roof, stacks against the house.  Within a day or so, heat from house melts a finger sized crevasse and the snow is no longer touching the house.

Assuming that there is reasonable siding on the house this is a non-worry item IMHO.

If however, the ground is impervious to water penetration, and you start getting a pool of water at the base of this crack, then watch that carefully that it doesn't get above the foundation course.  If this happens, you need to create drainage to take care of it.  I've not seen this happen in our climate.