Thai is not an Indo-European language.
Thai is a member of the Tai group of the Tai-Kadai language family. The Tai-Kadai languages are thought to have originated in what is now southern China, and some linguists have proposed links to the Austroasiatic, Austronesian, or Sino-Tibetan language families.
It is true though, that Thai through religion and cultural contact has borrowed many words from Sanskrit and Pali, which are Indo-European languages, and that บุรี buri is one of these words.
Richard W already discusses the same things I do below, in evidence of how the English word and the Thai word are not related, but here it is again:
The origins of the word borough/bury are discussed here:
http://www.etymonlin...hp?term=borough It comes from a Proto-Indo-European word, bhrgh, meaning 'high'.
The English cognate of 'buri' in Thai is found in 'police', 'policy' etc. The Proto Indo European root is *p(o)lH- according to the Online Etymology Dictionary.
See
http://www.etymonlin...php?term=policy It bears mentioning that Sanskrit is just one of many languages in the Indo-European language family, it is not the original Indo-European language, which is referred to as 'Proto Indo-European' and is a hypothesis based on analysis of the data available from the different branches of the language tree.
Sanskrit just happens to be the oldest one written down, which is why it is often used for reference and study purposes.
Scholars have been working on comparing words in Sanskrit and Latin and the oldest Germanic sources that exist, in order to see how phonological change has occurred. That way they have been able to establish some regular patterns by which the pronunciation of words have changed, and this is how they can present evidence for connections.