Bhutan is a remarkably safe destination, almost completely devoid of the scams, begging and theft that affects its neighbours. There are a couple of things to look out for, though.
Altitude It's unlikely you will have any problems with altitude unless you are trekking. Most of the places tourists visit lie below 3000m and the maximum elevation you can reach by road is around 3800m.
Aggressive dogs: Those scruffy dogs that sleep silently during the day turn into barking monsters at night. Bring earplugs. There is little danger of dog bites, but occasional rabies outbreaks occur in rural Bhutan, so be wary of big dogs guarding properties, especially if trekking.
Bad weather: Inclement weather can obscure the mountain views that you made such an effort to see and can affect Druk Air flights. Snow sometimes blocks the road temporarily on high passes in winter. In the monsoon season, heavy rain can turn trails and roads into a sea of mud and can wash away bridges.
Carsickness: If you venture east of Thimphu, you will spend hours driving on rough, winding roads and carsickness is common. Anti-motion medication such as Dramamine can help, but bring the anti-drowsy versions or you'll spend most of the spectacular drives snoring in the back seat.
Crime: Theft is still minimal in Bhutan, but as elsewhere it is growing along with the population, so you should be just as careful as you are in your home country.