According to Grammatist.com:
The term Far East usually refers to East Asia, including Mongolia, China, Korea, Japan, and Southeast Asia. It sometimes includes the Philippines, Indonesia, and the other Oceanic countries, and it sometimes includes eastern Russia and the Indian subcontinent.
The Near East is the eastern Mediterranean region once dominated by the Ottoman Empire.
Note: the origin of the term is of course in Europe, thus the "near/far" distinction.
- originally referred to everything between the other two Easts (Mesopotamia to Burma)
- but it now usually denotes the Near East in addition to Afghanistan, Iran, and the Arabian peninsula.
As a more authoritative example, according to the AP Style Guide (src),
the countries of the Middle East are Iran, Iraq, Israel, Kuwait, Jordan, Lebanon, Oman, Bahrain, Qatar, Saudi Arabia, Syria, United Arab Emirates, Yemen, Egypt, Sudan, and “the eastern part of Turkey.”
Noting that popular usage once distinguished between the “Near East” and the “Middle East,” AP recommends using Middle East unless a story source uses the term “Near East. ”