At the time of my visits, there were four open passes for tourists between Thailand and Myanmar, and three of them offered a stamping in your passport, thus offering a comfortable way to renew your Thai visa. Usually called a "Visa Run," the process consists of approaching one of these passes, checking out at the Thai immigration, getting entry and exit stamps from immigration at Myanmar, and then returning to Thailand, ready for a new 30-day "visa-on-arrival."
From north to south, the passes are:
Tachilek is in front of Mae Sai, the northernmost town of Thailand. By far, this is the most popular and orderly cross to Myanmar. Mainly Thais in search of cheap merchandises at the huge Tachilek market use the facilities here, but if you happen to be at Chiang Rai or Chiang Mai, it will be a very handy cross for you. From these cities, take a bus to Mae Sai, just a 1-hour trip from Chiang Rai, and from the Mae Sai bus terminal, take a 5-baht truck straight to the border, which is at the northern tip of the town. The Thai immigration office is about 1km before the bridge leading to Tachilek.
Myawaddy is in front of Mae Sot, allegedly the westernmost town of Thailand, and offers a more desolated option to be used mainly if you are touring the lower north of Thailand. It is conveniently close to Sukhothai. You must reach first the town of Tak – most buses travelling between Chiang Mai and Bangkok will leave you there or in the nearby Sukhothai. From there you have 1 hour by truck till Mae Sot, which, similar to Mae Sai, offers cheap transport to its western tip, the border with Myanmar. The Thai immigration office is just before the bridge connecting the countries, at the northern side of the road.
The Three Pagodas Pass is the closest pass to Bangkok, just further west from Kanchanaburi, but unfortunately you cannot get a visa there. From Kanchanaburi, you take another bus to Sangkhlaburi for around $2, from where is possible to cross the Three Pagodas border cross, Phra Chedi Sam Ong, to Myanmar, paying $10 to the Burmese immigration, but it is impossible to get a visa there.
Victoria Point (Kawthoung), in front of Ranong, the southernmost cross, involves a rather pricey travel in a boat between the two countries. The Thai immigration is at Saphan Pla (bridge fish) road, 500m before the port, and it is open from 8:30am to 4:30pm. A Son Tao takes some 20 minutes from the town to the pier and costs 7B. After getting an exit stamp at immigration, it is possible to get a half an hour ride by boat to Victoria Point in Burma for 300B. Once there, the Burmese will take a crispy $5 note for a fresh stamp on your passport. In Burma, it is possible to stay a few days at Kauw Thauong. At The Andaman club pier in Ranong, it is possible to get an exit stamp. Every day from 8am, there are boats to the club in Koh Son, and when coming back to the pier, an entry visa is issued. If less than 5 days are left, there's an extra 750B surcharge (looks like pure bribe, and probably the "club" is in a Thai island). The pier is some 20km north of the town.