No, generally Thai food doesn’t contain any dairy at all. One reason for that is because cows are very limited in Thailand, and therefore all dairy products are expensive here.
Even coffee in Thailand usually does not contain cream or milk. Instead, they use coconut milk as a dairy substitute. This is true for many Thai dishes as well.
You can always specify or ask when ordering Thai food to make sure that it doesn’t contain dairy. But generally, Thai food doesn’t contain any dairy and it’s not going to be a problem for you.
Eggs are another story, however. Eggs are commonly used in Thai dishes and fried eggs are often placed on top of Thai foods, so be cognizant of that if it is important to you.
Cheese is not commonly used in Thai food. Cheese is an expensive luxury in Thailand and is therefore not commonly incorporated in food dishes.