1. J.F. Hurley in a series of papers studied Lie algebras obtained by taking the multiplication table (with integer coefficients, due to Chevalley) of simple Lie algebras of classical or exceptional type and considering them over a commutative ring. The results describe center, ideal structure, etc. of such algebras in terms of the underlying ring. See, for example:  Ideals in Chevalley algebras, <a href="http://www.jstor.org/stable/1994801">Trans. Amer. Math. Soc. 137 (1969), 245-258</a>; Composition series in Chevalley algebras <a href="http://projecteuclid.org/euclid.pjm/1102977369">Pacific J. Math. 32 (1970), 429-434</a>; Centers of Chevalley algebras, <a href="http://www.journalarchive.jst.go.jp/english/jnlabstract_en.php?cdjournal=jmath1948&cdvol=34&noissue=2&startpage=219">J. Math. Soc. Japan 34 (1982), No.2, 219-222</a>. In the joint paper with J. Morita (Affine Chevalley algebras, <a href="http://dx.doi.org/10.1016/0021-8693(81)90299-4">J. Algebra 72 (1981), N2, 359-373</a>) he does something similar for some Kac--Moody algebras.

2. Some questions in free Lie algebras were considered over commutative rings, for example: D.Z. Djokovic, On some inner derivations of free Lie algebras over commutative rings, J. Algebra 119 (1988), 233-245, where centralizers of a member of a free generating set are studied. The latter reference is more or less random, probably more can be found in some books (Reutenauer?). 

There are more instances of considering Lie algebras over commutative rings (for example, plenty of papers about automorphisms of some trianlgular or close to them algebras), but, unlike in the case of Lie algebras over fields, all these are some isolated examples, rather than a coherent theory. The book(s) of Bourbaki recommended by Anatoly Kochubei are, probably, interesting in that regard. Bourbaki tend to state things in the utmost generality, and it is educational to see how quickly they have to give up considering Lie algebras over rings and have to "throw around properties of vector spaces" (quoting Darij Grinberg).

Perhaps the question could be augmented slightly by asking what is the reason for the absence of such a theory for Lie algebras (as opposed, for example, for associative algebras). Perhaps this is related somehow to the fact that classifying some natural (e.g., simple) classes of associative algebras is easier then that of Lie algebras (e.g., root space decomposition technique for Lie algebras which works over algebraically closed fields vs. "idempotent" technique for associative algebras which works over arbitrary fields and even over rings), but I venture into a sheer speculation here.