Unless posted (the moat around the Old City in Chiang Mai and markets are notable exceptions), you can legally carry an open drink on the street in most places throughout Thailand.
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While you may think that holding your open container in a brown paper bag will protect you from open container laws, it does not. It also does not protect you from being charged with public intoxication.
Drinking alcohol in Thailand is legal, but it is restricted for use in such locations as temples, parks, gas stations and other. It is allowed to drink alcohol in public, but walking around with an opened bottle in Thailand is less socially acceptable.