Traditionally, it should be "no one but him" if the phrase is an object phrase, and "no one but he" if it is a subject phrase. Consider the following passages from Shakespeare:
It is, because no one should sway but he;
No one but he should be about the king;
I'll fight with none but thee;
for I do hate thee
The quarrel toucheth none but us alone;
Then yield, my lords; and here conclude with me
That Margaret shall be queen, and none but she.
You use the same preposition you would use if you removed "none but"—he should sway; he should be about the king; I'll fight with thee; The quarrel toucheth us; she [Margaret] shall be queen. This is explained quite cogently in Lindley Murray's English Grammar from 1808.
Today, some people still use this system, although it appears that the majority of people always use the accusative case, as explained in Barrie's answer.
Garner's Modern American Usage recommends the nominative case when the but phrase precedes the verb, and the accusative when it follows. They give an example:
None of the defendants but he were convicted.
None of the defendants were convicted but him.
From the American Heritage Book of English Usage:
Should you say No one but I read the book or No one but me read the book? If but is a conjunction in these sentences, you should use the nominative form I. If but is a preposition, you should use me. ...
These recommendations [the accusative case] are supported by 73% of the Usage Panel when the but phrase precedes the verb and by 93% of the Usage Panel when the but phrase follows the verb.
So when but comes before the verb, it appears that around a quarter of the Usage Panel is still following the traditional system, which has been used since the time of Shakespeare.
Which one should you use on a standardized English test? They shouldn't put this kind of question on a standardized test, since it appears that grammar books differ in their recommendations. But it appears that they do. I don't know which grammar the testing agencies like.