COPENHAGEN—This week’s stream of refugees into Denmark has sent tremors through this Nordic country’s politics, as the arrival of hundreds of asylum seekers came up against the government’s anti-immigration policies.

The tension has left conservative Prime Minister Lars Lokke Rasmussen in a political bind. His right-wing allies say he has failed to tighten borders, while his left-wing opponents say he has been too tough and has embarrassed the country.

“This...