Coxeter groups form a very large class of groups defined by generators and relations, whose Poincare series are unlikely to have a common geometric interpretation. However, the Poincare series of an affine Weyl group has been important in work of Bott on topology of Lie groups, as well as in work of Iwahori-Matsumoto and others on p-adic groups, etc. References are given in section 8.9 of my 1990 Cambridge Press book Reflection Groups and Coxeter Groups, while the seminal 1965 IHES paper by Iwahori-Matsumoto is available online at www.numdam.org (search for article):
Iwahori, Nagayoshi; Matsumoto, Hideya,
On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups. Publications Mathématiques de l'IHÉS, 25 (1965), p. 5-48
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There are Kac-Moody groups in some generality, with associated "Weyl groups" and
"Bruhat decompositions"; but Bruhat cells may have finite dimension in some cases, finite codimension in others. It gets complicated.
It's possible that work of Michael Davis and Ruth Charney on hyperbolic Coxeter groups would be relevant to your question, but I don't know enough about that.