REYKJAVIK, Iceland—Iceland’s tourism push helped save the once-remote island from a deep economic crisis, but now that business is booming the North Atlantic nation is straining under the weight of the windfall.

Attracted by its spectacular volcanic landscape and easy air routes, about 2.2 million tourists are expected to flow into the country of 330,000 people this year, almost five times as many as in 2010, according to the International Monetary Fund. Tourism is now the country’s biggest industry, ahead of its traditional...