On the whole, tourists are not at a great level of risk of any danger from the political situation if they stick to the regions that have been opened up. It's far more likely that if you do have any issues they will be related to the usual problems with hygiene and safety standards that are seen in developing countries. We'd advise that you make sure to have full and comprehensive travel insurance in place in case anything does occur. This is another reason to stick to areas that welcome tourism as you might not be covered if you venture into areas that your government considers unsafe.
Scams in Myanmar
In terms of being hassled or scammed by locals, the chances of being impacted by these sorts of things is fairly low. In the cities and Yangon in particular you may be offered money changing services on the street which should be avoided as you'll usually find out that you've lost a deal of money in the process! Generally though a polite ‘no thank’ you is more than enough to send anyone bothering you on their way. Problems like robbery and petty crimes against tourists are far lower than many other countries: just be sure to keep your wits about you and be sensible.
Road safety issues
The main issue that's likely to be encountered are unsafe roads, dodgy pavements and poor driving standards.
The roads can be little more than dirt tracks in some places and even those in better repair are often full of pot holes and not well maintained. Combine this with old and poorly maintained vehicles and this is the best chance of getting to an accident.
Always be very careful if driving or cycling and many choose to avoid night buses for safety reasons. Pavements are often uneven and littered with holes, even in the cities, so be really careful when walking around. At night try to make sure you've got a light so you can see where you're going as roads are often poorly lit.
The next danger is getting sick. We and pretty much everyone we met had at least one bout of sickness in Myanmar, although luckily nothing serious! Don't drink tap water under any circumstances and be careful to eat at reputable places. If you're going to risk the street food try to stick to busy places and make sure any hot food you eat is piping hot throughout.
Rabies and other diseases spread by insects and animals are another issue. Rabies is fairly well spread amongst dogs and monkeys in particular so do avoid as far as you can as it's not worth the risk.
Make sure you see a travel doctor before visiting so you know what vaccinations you need and take precautions while you visit. Many people take anti-malarials and mosquitos can also spread dengue and other illnesses.
Finally, if anything does happen the standards of medical care in Myanmar can be quite basic. Especially when you are outside the cities facilities can be poor and many people will not be able to speak English to assist you.
If anything serious occurs you might find that you will be evacuated to hospitals in Thailand for a far higher standard of treatment.
Whilst instances of foreign visitors being injured or getting seriously ill are fairly low, it's still a destination where it's worth taking all reasonable precautions due to the issues with healthcare.