How does the Runesmith class interact with the Snowcasting feat from Frostburn? Does the Eschew Materials feat change this?
They don’t interact at all. Snowcasting can be used with runes the same as it can be used with any other spells; it being cast from a rune doesn’t change anything. Spells can have multiple material components, so having the spell require both the rune and some snow or ice for Snowcasting isn’t any kind of a problem. You can still use Snowcasting’s option for picking up snow (by taking longer to cast the spell) along with the rune: that works exactly like it would for a prepared spell.
As for Eschew Materials, again, everything works exactly the same as it would without rune magic. Eschew Materials will not eliminate the requirement for having the rune, but can be used on any other material components as normal.
Now, how Eschew Materials interacts with Snowcasting—whether rune magic is involved or not—is somewhat controversial. Rules as written, when you use Snowcasting you add a new material component to the spell—snow or ice—and since that component costs nothing, Eschew Materials allows you to ignore that component. A lot of people dislike this, arguing that if you use Eschew Materials to eliminate the snow or ice, then you aren’t using snow or ice and therefore not triggering Snowcasting. Unfortunately for them, that’s not how Snowcasting is written, and honestly, Snowcasting is a combo-enabler—it does absolutely nothing on its own—and having to use yet another feat in the combo is a fairly significant cost.
But rune magic changes nothing about this situation.