Can I apply for tourist visa while living in a different country as an expat?

You should apply for a Schengen visa at the consulate for your place of residence. This rule was made because the local consulate is best suited to judge your living conditions and your likelihood of being a genuine tourist or business traveler.

So yes, for Schengen it is explicitly permitted. As Jan points out, they will take your ties to your place of residence into account, which may be less than for a citizen.

As others have explained at length, it depends on the rules of the destination country and the nuances of your situation. But a more direct answer is possible:

• The Schengen area has a rule that you must apply in your place of residence. It means that if you are living in Thailand, it may even technically be mandatory to apply from there and nowhere else. On the other hand if you cannot prove that you are a resident in Thailand (the consulate will decide what document they require for that), it will be difficult to apply for a Schengen visa there (even if there are limited exceptions to this rule).
• The UK doesn't care, you can apply at any visa application center, irrespective of your residence status. They warn that they may keep your passport during processing (which may be especially inconvenient in a foreign country) and that it may take longer to process visas for non-residents.

The short (and frankly not very helpful) answer is that it depends on the policies of the country issuing the visa, i.e. the country you intend to visit. Each country will have its own conditions specifying where you must apply from, who is eligible to apply, etc. This is complicated addionally by the fact that ‘nonimmigrant visa’ is a very broad category that are unlikely to be used by the country you are trying to visit in determining whether or not you can apply and if so which embassy/consulate you would have to apply from.

However, in my experience there are some general guidelines that you can probably expect to be accurate (you should, however, always double check!):

• If your non-immigrant visa for Thailand is a tourism or business visa (typically a short-stay visa of three to six months validity), you are likely not considered a resident of Thailand and would have to apply from wherever your residence was before (likely India).

• If your non-immigrant visa is a medium or long term visa (essentially a year or longer) and you move your place of residence to Thailand (I don’t know whether Thailand has residency registration policies as e.g. Japan or Germany have; but even if not, if you e.g. notify your bank of your new Thailand address, open a bank account in Thailand, etc. that should count), then you will likely have to apply as a resident of Thailand.

• Exceptions may apply. The top-of-my-head case would be someone travelling the world where it would not make sense to apply for visas for the entire trip from back at home; in certain cases they may be permitted to apply for a third country’s visa from a country they are not resident of. Exceptions may or may not be granted.

As you are using the term expat, I assume the second bullet point will apply to you: you have a working contract in Thailand for several months, would relocate to Thailand even if not as an immigrant and open a Thai bank account to receive your salary in Thai currency. That case would be similar to mine when I worked in Japan as a postdoc. My (non-immigrant) visa was granted one year at a time (which was not in line with my work contracts) and had to be extended; the assumption was I was going to leave Japan at some point (it proved true). Nevertheless, I applied for a Chinese visa through the consulate in Osaka rather than through the consulate in my country of citizenship as I was considered a resident of Japan.

A final word of warning in your specific case: as an Indian national applying for visas to enter the UK or Schengen you will face a significant burden of proof to show that you are indeed a genuine visitor, i.e. that you have sufficient ties back to, here, Thailand to ensure you will return. Living in a country that is not that of your citizenship may make the application process more difficult. It is certainly not impossible – for example, I know a Nepalese citizen, Japanese resident who nevertheless managed to acquire a Schengen tourist visa to attend a scientific conference in Germany – but you may have to jump through additional hoops and provide a more convincing case to the consulate. A longer term working contract and a longer time having spent in Thailand will probably work to your advantage.