Is there a linear characterization of being the inverse of a Stieltjes matrix? In other words, if $A$ is a $n \times n$ matrix over the reals, is there a set of linear equations in the entries of $A$ such that $A$ is a the inverse of a Stieltjes matrix if and only if these linear conditions are satisfied?
As far as I know, an exact characterization of the form you want is unknown. A necessary condition, for an inverse M-matrix (weaker than inverse Stieltjes) is the so-called "path product condition" - see http://www.math.temple.edu/~abed/JS07.pdf.
Another necessary condition is that the principal minors be all positive (and they are multilinear in the entries): see for example: inverse m-matrix
ADDITION: There is an "if and only if" characterization of the sort you want for inverse M-matrices (well, almost, since it's multilinear, but maybe that's what you meant :). See Theorem 2.9.1 in the new survey by Johnson & Smith:
http://www.sciencedirect.com/science/article/pii/S0024379511001273 Charles R. Johnson & Ronald L. Smith, Inverse M-matrices, II, Linear Algebra and its Applications, Volume 435, Issue 5, 1 September 2011, Pages 953–983
Let $A \geq 0$. $A$ is an inverse $M$-matrix iff:
(a) $A$ has at least one diagonal positive entry
(b) all Schur complements of order 2 are nonnegative
(c) all Schur complements of order 1 are positive
A partial answer is given in this paper.:
￼A linear algebra proof that the inverse of a strictly ultra metric matrix is a strictly diagonally dominant Stieltjes matrix (Nabben + Varga, SIAM journal of matrix analysis, 1994).