Political tensions stoked by the European Union’s migration crisis have flared in Poland, causing what has been one of the most influential members in the bloc’s east to take a more confrontational stance with Brussels.

Polish opposition to EU plans to spread the burden of taking in migrants across the bloc helped the right-wing nationalist Law and Justice party clinch an overwhelming victory in late October, ending eight years of rule by the centrist, pro-EU Civic Platform, which had earlier agreed to accept several thousand...