“If I could put all my money into Myanmar, I would,” veteran U.S. investor Jim Rogers declared in 2012. Rogers, who co-founded the Quantum Fund with hedge fund billionaire George Soros, has lived in Asia for many years and knows a bargain when he sees one.

When Myanmar opened up to the world after decades of military rule, hopes were high that democracy would be restored and the economy invigorated. From tourism to telecoms, business opportunities abound as industries that had long languished in the absence of private competition were roused out of their slumber.

Local consumption soared amid a thriving economy, with the Burmese now plugged into global trade, wearing Nike, drinking Coke and surfing smartphones and — along with a growing expat community — spurring demand for higher-end real estate.

There have been many economic success stories since the early days of the country’s re-opening for business, and ...