Is every poset the poset of prime ideals of a ring?

Hochster answered this question in his thesis (by finding such conditions). See: Hochster, M. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 1969 43–60.

Beside considering the article by Hochster you could also take a look at

WILLIAM J. LEWIS, The Spectrum of a Ring as a Partially Ordered Set, JOURNAL OF ALGEBRA 25, 419-434 (1973).

It is online:

www.maths.manchester.ac.uk/.../Lewis-Thespectrumofaringasapartiallyorderedset.pdf