When is it expected to write someone's name correctly? It is generally expected that European names be spelled correctly in academic correspondence and citations. This is expected, even when they use symbols not in the English language, when publishing in English (provided that a modern typesetting system will support it).

For instance, the German name "Müller" is expected to be spelled correctly, even though it could be spelled "Mueller" in the English Alphabet (without loss of information). Similarly, Gaelic names such as "Ó Ceallaigh" would not be Anglicised, despite a long history of this having been done before.

However, this is not the case for Asian languages, even Japanese names where they do not typically have the culture of adopting a nickname in Western Countries. For example, "田中" would be spelled as "Tanaka" to confirm with English language readers, despite Japan having it's own phonetic conventions to give the desired reading (e.g., "タナカ").

For a more comparable example, why is it ok to spell “Tokyo” in English when “Muller” is incorrect. The correct romanisation of 東京 is Tōkyō. Yet this is not used, nor is Toukyou or とうきょう which would both be more accurate. This misspelling occurs for names of people in Asian languages as well as names of places.

It's typically argued that this is because English-speaking audiences could not read Japanese names in their writing system but the same could be said for the umlaut, which is often mispronounced or misused (e.g., Mötley Crüe). With digital typesetting systems, would be entirely possible to spell "田中" correctly in a citation, even by someone who cannot understand the meaning, just as we do for diacritics for European names.

If spelling someone's name correctly is a matter of respect, when is it necessary to do so and why are there exceptions to this?
I think "correctly" is the wrong word to use here, and it will cloud the issue. What's "correct" for a name is somewhat ill-defined (I've known a Risa and a Lisa who both had the same Japanese name. Which of them was incorrect?). The real standard is to spell in accordance with the person's wishes.

As a German person, the blanket assertion "the German name "Müller" is expected to be spelled correctly" is not in my experience true. Rather I would say, as the ease of writing the umlaut electronically increases, the likelihood that someone will be unhappy that you didn't bother to make the minimal effort increases too. There are still plenty of systems that only take ascii characters and people will make do with Mueller or even (sigh) Muller. Japanese people know quite well that you probably couldn't read "田中" nor "タナカ", and they will themselves sign their correspondence to you "Tanaka". They won't expect you to make the effort to learn how to read and type kanji, because they know it's exceptionally more difficult than finding a diacritic. (They will however be quite happy if you do manage.)

For citations, the issue is a little different: it is expected that you cite the authors as written on the paper. If you are citing an English-language paper, it is very likely that you will be citing "Ryouichi Tanaka and Robert Müller", because that's what they called themselves. If you are citing a Japanese manuscript, you are expected to cite "田中良一　ロバート・ミュラー". Maybe as "田中良一　ロバート・ミュラー (Ryouichi Tanaka and Robert Müller)" if you are yourself writing in English.

To sum up: 


When is it expected to spell someone's name correctly? Always, but correctly ≠ in the original alphabet, or following some specific romanization rules.
When is it expected to spell someone's name in the original alphabet? When you can reasonably do so without major effort, and it does not hinder communication.