Republic of China / Taiwan is the most obvious example of a place that is not diplomatically recognized by most countries (although trade offices and the like typically handle things for travelers that embassies and consulates do).

There are also places like Hong Kong, which is a political subdivision of country (People's Republic of China in this case), but has separate immigration procedures from the country that it is part of (and crossing between it and the rest of the country that it is part of requires going through immigration procedures).