Bruneians are generally very tolerant and will understand that visitors are not familiar with all customs and Islamic traditions. Nevertheless, keeping these few things in mind will go far in enriching your experience and will show the Bruneian people that you respect and appreciate their culture:
Tourists should observe the local dress code and dress modestly. Clothing comfortable for hot weather is acceptable, except when visiting places of worship or for social and business functions.
Bruneians greet each other by lightly touching hands and then bringing the hand to the chest. Some people do not to shake hands with members of the opposite sex.
You should not point with your fore finger; instead, use the thumb of your right hand with the four fingers folded beneath it.
When visiting a mosque, all visitors should remove their shoes. Women should cover their heads and not have their legs or arms exposed. You should not pass in front of a person at prayer or touch the Koran.
Gifts (particularly food) should only be passed with the right hand, although it is acceptable to use the left hand to support the right wrist.
It is polite to accept even just a little food and drink when offered. When refusing anything offered, it is polite to touch the plate lightly with the right hand.
During the Islamic fasting month of Ramadan, Muslims do not take food from sunrise to sundown. It would be inconsiderate to eat or drink in their presence during this period.
In deference to the Muslim majority, alcohol is not sold in Brunei, but private consumption by non-Muslims is allowed. Non-Muslim tourists are allowed a generous duty-free allowance of 2 bottles of alcohol (wine, spirits etc) and 12 cans of beer per entry, and may consume alcohol with sensible discretion in hotels and some restaurants.
Be sensitive about topics such as religion and the royal family.