In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group of finite rank and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard basis of this algebra to a more canonical basis, with the change of basis coefficients involving polynomials indexed by pairs of elements of the Coxeter group (in the Bruhat ordering) over $\mathbb{Z}$. Even though the evidence at the time was quite limited, they conjectured following the statement of their Theorem 1.1 that the coefficients of these polynomials should always be non-negative. (In very special cases this is true because the coefficients give dimensions of certain cohomology groups.)

Several decades later, Wolfgang Soergel worked out a coherent strategy for proving the non-negativity conjecture, in his paper Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln uber Polynomringen. J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525. This is posted on the arXiv here. Now that his program seems to have been completed, it is natural to renew the question in the header:

What if any implications would the non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials have?

It has to be emphasized that in Soergel's formulation and the following work, the non-negativity is not itself the main objective. Instead the combinatorial framework proposed was meant to provide a more self-contained conceptual setting for proof of the original Kazhdan-Lusztig conjecture on Verma module multipliities for a semisimple Lie algebra (soon a theorem) and further theorems in representation theory of a similar flavor. But Coxeter groups form a vast general class of groups given by generators and relations, so it is surprising to encounter such strong constraints on the polynomials occurring in this generality.