computational geometry - What is the most general class of metric spaces for which the closest pair of points in a finite subset can be found in time O(n^(1+eps))?

I assume you are aware of the classic paper by Jon Bentley,
"Multidimensional divide-and-conquer"
[Commun. ACM 23(4):214-229 (1980)],
in which he showed how to find the closest pair of points in $mathbb{R}^3$
in the Euclidean metric in $O(n log n)$ time.
His algorithm works in arbitrary dimensions in $O(n log^{d-1} n)$.
I realize I am not answering your question about metric spaces, but it might be worth revisiting
his algorithm to see how heavily it leans on the norm.

Rabin's 1976 randomized algorithm achieves $O(n)$ expected time.
An updated detailed analysis is in the paper
"A Reliable Randomized Algorithm for the Closest-Pair Problem"
by Martin Dietzfelbinger, Torben Hagerup, Jyrki Katajainen, and Martti Penttonen
[Journal of Algorithms 25(1): 19-51 (1997)].
Again I am not addressing your focus on other metric spaces, but these efficient algorithms
for Euclidean distance would be a place to start.