While the possession of a non-immigrant visa is no longer a requisite for foreigners to purchase and register a motor vehicle in Thailand, it still is the case for application for a Thai driver's license.
Now some supplementary information:
Your Embassy in Thailand regularly issues “address certificate” for the purpose of purchasing a motor vehicle or application for Thai driver's license. This piece of paper is pretty expensive, ranging from 700B to 1000B depending on the Embassy, and it may take a couple of days.
An alternative is to get an address certificate from the Immigration Office. This one is written in Thai and costs 500B. In case you have been negligent of the 90-day address report to the Immigration Office, you may want to stay away from there.
There are other alternatives, like presenting your work permit to prove your address, or asking a local police station to issue one, but these methods are rather irregular and may or may not work out.
If you have a valid International Driving Permit or a driver's license from another country, you don't need to take the written test or the driving test.
The International Driving Permit makes things easiest. You don't need to have it certified or translated. Just bring it to the Land Transport Office.
In case you don't have an International Driving Permit, but have a driver's license from another country,
This is largely a formality. The certificate can be obtained from any hospital or clinic, its fee ranging from 20B to 200B depending on which hospital or clinic you choose. The certificate certifies that you are not suffering from certain physical or psychiatric disorders specified in the law, but unless you specifically claim that you are suffering from such a disorder, the doctor will certify that all is well with you.
The Motor Vehicle Act of Thailand specifies ten categories of driver's license (eg. car, motorcycle, tuk-tuk, etc.). You must apply for them separately and they issue separate driver's licenses. If you intend to apply for both car driver's license and motorcycle driver's license at the same time, you only need one original document for each of the above-mentioned certificates, and make a set of photocopies for application of additional driver's license.
According to the Motor Vehicle Act of Thailand, what you get for the first year, both for the car and motorcycle, is a so-called “temporary driver's license” or “provisional driver's license.” Its color is blue. From the second year on (at your first renewal), you get a full driver's license. Its color is green.
It is not clear what is temporary or provisional about the driver's license for the first year. Mostly, it works as a full driver's license. One thing is, you cannot apply for an International Driving Permit in Thailand with a temporary driver's license. Another thing is, as I have heard, that some insurance companies put restrictions on its comprehensive insurance policy (but not compulsory insurance policy) for drivers with a temporary driver's license.
At the moment, a Thai driver's license issued to a foreigner is only valid for one year, unless he has an immigrant-visa (permanent resident status). You are supposed to renew your driver's license before it expires, but not earlier than one month before it expires. You can still renew your driver's license after it expires, and it has been a myth that officials at the Land Transport Office tell you to renew your driver's license after it expires, but it is against the law to drive a motor vehicle with an expired driver's license.
For driver's license renewal, you don't need the address certificate (unless you have changed your address) or health certificate, but you still need to be in possession of a non-immigrant visa. In some provincial Land Transport Offices, officials may ask you to produce address certificate upon driver's license renewal. I'm not sure about such a case, whether those officials have a certain discretion as to what documents are required for driver's license renewal, or they're simply ignorant of laws and regulations. Probably the latter.
Many countries issue different levels of motorcycle driver's license, imposing a certain restriction to the size of the engine allowed for each level. Thailand issues only one level of motorcycle driver's license, though there is an age-based restriction (15 - 17 years old … motorcycles under 90cc).
When you go to the Land Transport Office for driver's license application, a tout may spot you and talk to you, offering to take care of the procedure for 1000B - 1500B. If you have all necessary papers, you don't need his assistance even if you don't speak a word of Thai. On the other hand, if you don't have necessary papers, he cannot help you at all.
There's another kind of personality who can actually help you get a Thai driver's license without necessary papers, but this kind of agent won't be hanging around in front of the Land Transport Office.
A Thai driver's license is an official document proving that you reside in Thailand, and is accepted in lieu of a passport in a lot of situations.
It also distinguishes you, as a resident foreigner, from visiting tourists. Notably, many National Park gatekeepers unofficially let you in for the Thai rate of 20B instead of the foreigner rate of 200B if you present your Thai driver's license.