VILSECK, Germany—For years, wherever the U.S. Army went, Burger King was sure to follow.

When the Army rolls into Poland next month to take positions near the Russian border, however, the creature comforts will stay behind. No Whoppers. No foot-long Subway sandwiches. No Pizza Hut pies or caramel lattes at the Army’s version of Starbucks, Green Beans.

“You...