"Remember thou shalt die": Shall/will

In Shakespeare, shall and will were not used according to the "traditional rule in Standard British English" described in your link. You can see from this Ngram that there was a big change in the rules for shall and will between 1600 and 1700, at least for first person. Since the comments say this line was composed around 1600, the grammar would presumably be that of Shakespeare.

As far as I can tell from looking at examples in Shakespeare, for people he used will to indicate volition on the part of the actor, and shall to indicate lack of volition. Thus, thou shalt die means you are dying without necessarily wanting to.

EDIT: I have found a reference for this usage. From The Cambridge History of the English Language:

It has been suggested (e.g. (Jespersen MEG IV 18.1; Strang 1970: 206) that the divided use of the two auxiliaries to indicate future time might go back to the model set by the Wycliffite Bible translation, which used shall for unmarked and will for volitionally marked future. This practice would have been copied by the schools in their translation exercises. This theory certainly gives a much simplified picture of the development; yet it seems that will developed its pure (predictive) future use later than shall, in colloquial speech, as a `change from below'.

The peculiar pattern of distribution in which shall is the future auxiliary used with the first-person subject while will is used in the second and third persons can be first traced in Early Modern English. The grammarian Mason first states this rule in 1622, and Wallis in 1635 (Visser §1433) but the tendency can be traced in texts as early as the 16th century.