I had exactly this issue a few weeks ago when I went backpacking in the Sierras. I wear non-GoreTex Montrail Hardrocks and usually hike in the Mid-Atlantic where trails aren't too dusty/sandy. My feet got very grimy in the toe box area and I ended up getting blisters between my toes (which I've never had before).
I still like my shoes for the breathability and ability to walk across streams without changing shoes and have them dry quickly, but next time in the Sierras I'll tape my toes before I start. What I'd really like is something like my Dirty Girl gaiters that covers the entire front of the shoe, not just the rear half.