This morning I went ahead and changed the app to pull from MapQuest's public OpenStreetMap-based tile servers, with appropriate tweak to the credits line. This will put less usage on the primary OpenStreetMap public tile servers, as we don't yet have a good idea just how much traffic we'll end up driving.
People using nightly builds may notice that the map style is slightly different, but should work the same and have the same content. People using the current Android and iOS app releases will not see any difference at this time; those continue to use Google Maps.
We're in the process of getting things set up to run our own OSM tile servers, which we'll eventually use on Wikipedia itself as well as in the mobile app, but I'm not sure what the schedule on that still looks like.
OSM's tile usage policy: https://wiki.openstreetmap.org/wiki/Tile_usage_policy
Docs on the MapQuest tile servers: https://wiki.openstreetmap.org/wiki/Mapquest#MapQuest-hosted_map_tiles
-- brion