Are there any areas of writing, literature or science where æ or œ are still used? Are there contexts where they are still considered mandatory?
I'd say that history and linguistics are two of them.
In the area of history, you will probably be better understood if you spell "Alfred the Great" instead of "Ælfrēd" - especially if you write for an audience of non specialists. Although, this is a matter of personal taste; I prefer to see "Ælfric" rather than "Alfric of Eynsham", because having to deal with several spellings for the same name means more effort for the reader.
However, the use of the "Æ" is probably not mandatory in history.
In linguistics however things are different.
In studies pertaining to Old English, the use of the "Æ" is expected.
Besides, Old English modern spelling conventions has a number of special characters (ð, þ being the most common) which makes "æ (short)" or "ǣ (long)" just one of many typographical peculiarities.
In this case, trying to avoid the hassle of key sequences is the best way to make yourself unintelligible.
In addition, as JSBangs rightfully points out in his comment, the "æ" is one of the phonetic symbols needed to represent the sounds of the English language in the International Phonetic Alphabet.
In which case it is also mandatory.
There are tons of œ ligatures in French like sœur (sister) and cœur (heart.)
What I've noticed is that in British English and American English some words are spelled differently, which leads me to believe that there was a ligature, as confirmed by Wikipedia (just now as you've sparked my interest on the subject).
For example, mediæval is now just spelled mediaeval in Britain and medieval in the States; encyclopædia is another one. One example of an œ ligature in English would be fœtus (or fetus in American English).
I don't know if there are any known ligatures that exist in English today, but they are still common in French.
I would suggest that computerisation has caused the end of ligatures. It's much easier just to type ae than bind a keyboard shortcut to the ligature version.