Upon Andre's request, I am rewriting my comments as an answer, even though I am on a somewhat shaky ground since my experience with infinite-dimensional algebraic geometry is very limited.

It seems that for "most" metric spaces (and hyperbolic space is one of the "most") distance functions (or their squares) will span an infinite-dimensional real vector space. I checked this in the case of distance functions on the hyperbolic space (of dimension $\ge 2$, of course) and the same should hold for the distances squared. The point is that for every metric space $X$, the map $i=i_X: x\mapsto d_x$, where $d_x: X\to {\mathbb R}$ is the function $d_x(y)=d(x,y)$, determines an isometric embedding from $X$ to a linear subspace $V$ in $C(X)$ (space of continuous functions on $X$) with the sup-norm on $V$, where $V$ is the linear span of the image of $i$. On the other hand, hyperbolic space (of dimension $\ge 2$) does not embed isometrically in any finite-dimensional Banach space. Thus, doing algebraic geometry on the hyperbolic space becomes (in my mind) a somewhat daunting task since you have to consider the ring of polynomial functions ${\mathbb R}[V]$ on the infinite-dimensional vector space $V$. In particular, you loose the Noetherian property which makes life difficult. The only context where I have seen infinite-dimensional algebraic geometry is the *ind-schemes.* Ind-schemes appear naturally when one works with, say, affine Grassmannians and which I had to do exactly once in my life (I mean, thinking of affine buildings in algebro-geometric terms). As far as I can tell, dealing with the ind-scheme based on the ${\mathbb R}[V]$ for the hyperbolic space, would amount to considering geometry of finite configurations of points in ${\mathbb H}^n$ (and, occasionally, lines). I could be mistaken, but for the union $Y$ of two distinct geodesic segments in the hyperbolic space, linear span of the image of $i_Y$ (where we use the restriction of the distance function from ${\mathbb H}^n$) will be infinite-dimensional, so you are not allowed to use more than one geodesic. While geometry of finite subsets of hyperbolic spaces has some uses (see the discussion of the sets $K_m$ below), it strikes me (I am a hyperbolic geometer) as mostly boring. (Maybe logicians can add something interesting here since considering finite subsets in ${\mathbb H}^n$ we are dealing with the elementary theory of the hyperbolic space.)

Gromov (see e.g. his book "Metric Structures for Riemannian and Non-Riemannian Spaces") introduces, for every metric space $X$, the collection of subsets $K_m(X)\subset {\mathbb R}^N$, $N=\frac{m(m-1)}{2}$. The set $K_m(X)$ consists of $N$-tuples of pairwise distances between various $m$-tuples of points in $X$. Then $K_3(X)$ (under some mild assumptions on $X$, e.g., $X$ is an unbounded geodesic metric space) is defined by triangle inequalities and nothing else. The set $K_4(X)$ is quite interesting, since all the "curvature" conditions on metric spaces are defined via quadruples of points. However, Gromov could not come up with any interesting uses for $K_m(X), m\ge 5$, and I do not know what to make of these sets either. Thus, the ind-scheme approach to the algebraic geometry of the hyperbolic space might yield something geometrically interesting for quadruples of points, beyond which algebraic geometry is likely to get disconnected from (geo)metric geometry.

Lastly, MO discussion at Is there an algebraic approach to metric spaces? is related to David's question.