As a responsible tourist, you should  try to make sure that the people you meet enjoy your stay in their country as much as you do - or at least, that you do not offend your host. "Responsible Tourism Mekong" suggests the following guidelines for tourists:

    Do say “Mingalarbar” when meeting someone, use  “U” in front of men names and “Daw” in front of women names;
    Do let the oldest be served first;
    Do offer articles with both hands and keep both feet on the ground;
    Do bend slightly in front of the elders;
    Do dress and act decently, speak slowly and clearly;
    Do ask permission before taking photographs;
    Don’t touch anybody’s head; Don’t touch women;
    Don’t point a finger straight in the face;
    Don’t step over any part of the person;
    Don’t go where you are advised not to go;
    Don’t traffic, handle, or use narcotic drugs;
    When entering pagodas or monasteries, wear decent clothes (no shorts, bare shoulders or chests) and take off your shoes;
    Don’t sit with your back towards Buddha’s image;
    Show respect to monks, novices and nuns, don’t offer to shake hands, a woman should not touch a monk, don’t step on a monk’s shadow;
    Don’t handle Buddha images or sacred objects with disrespect or keep them in inappropriate places (on the floor for example)

The Ministry of Hotels & Tourism Myanmar proposes the following Do's and Don'ts:

Typical Character
    Friendly, helpful, honest, but proud.
    Treat everyone with respect and you will be respected.
     
Speaking
    When addressing people, don’t leave out U (which stand for Mr) or Daw (which stand for Ms/Mrs)
    Speak slowly and clearly.
     
Manners
    Not always necessary to shake hands.
    Don’t hug or kiss in public.
    Don’t touch any adult on the head.
    Don’t step over any part of a person, as it is considered rude.
    Accept or give things with your right hand.
    In Myanmar, unlike the Indian continent, nodding mean YES, and shaking head means NO.
     
Eating
    For hygiene reasons, eat only in decent restaurants.
    When not available, always eat heated food.
    Don’t eat food purchased from street vendors.
    Don’t drink tap water.
    Drink only bottled water and soft drinks that haven’t been opened yet.
    Let the oldest be served first.
    Chinese food is common and suggested.
    Myanmar food are often complained as ‘oily’.
    To try good Myanmar food, go to decent restaurants in Yangon area, where they cook Myanmar food according to international standards.
     
Shopping
    When buying gems, sculptures, or any expensive souvenir, make sure it comes with an export permit.
    Buy arts from authorized dealers only and get a certified receipt.
     
Safety
    Don’t leave expensive items in your room. Use safe deposit box.
    Beware of cheats, swindlers, imposters.
     
Health
    Stay away from narcotic drugs.
    Carry some medicines for diarrhea.
    If sick, don’t worry. All doctors are English literate.
    Health insurance is not available

Traveling
    Accept that facilities may not be the best.
    On trains, keep windows shut.
    Speed or distance descriptions are in miles, not kilometers.
    Carry toilet paper in your bag.
     
Visiting
    Most Myanmar do not wear shoes in their homes. Take off when visiting.
     
Moving About
    Don’t jay walk. Watch where you walk and what you step on.
    If driving, city speed limit is 30 mph. Drive on the right side.
     
Religion
    At religious places, remove footwear, but to remove headwear is not necessary.
    Avoid shouting or laughing.
    Avoid being a nuisance when taking photographs.
    Tread Buddha images with respect.
    Tuck away your feet. Don’t point it toward the pagoda or a monk.
    Don’t play loud music in these areas. Note that Buddhist monks are not allowed to listen to music.
    Do not put Buddha statues or images on the floor or somewhere inappropriate.
    Don’t touch sacred objects with disrespect. Hold them in your right- hand, or with both hands.
    Leave a donation when possible.
    Show respect to monks, nuns, and novices (even if they are children).
    Don’t offer your hand to shake hands with a monk.
    Sit lower than a monk and elders.
    Don’t offer food to a monk, nun, or a novice after noon time.
    A woman should not touch a monk.

 

Credit to The Ministry of Hotels & Tourism Myanmar and Responsible Tourism Mekong.