I met a student at the JMM poster session last year who was showing a project studying the distance between organisms, using the tree metric on the evolutionary tree. He had shown some statistical data for tracking differences in traits against the organisms' distance on the tree, scaled logarithmically. It was really cool but unfortunately I lost the notes I took on it, including his name.
The underlying idea seems quite natural but I am unable to find any mention of the concept online, possibly because I am not typing the correct search words. To be clear, the evolutionary distance $d(A,B)$ from organism $A$ to organism $B$ would be computed as follows. Find the most recent common ancestor, then draw the unique path on the tree from $A$ to $B$ and passing through that common ancestor. The number of branches on that path is $d(A,B)$. For example, my distance to my mom is 1, to my brother is 2, to my first cousin is 4, and to a spider is extremely large.
Can you direct me to any work on this topic?
Unfortunately, I'm not sure about the exact work you're looking for (it's not my area exactly), but I'll try to give what comes to mind. Maybe asking at the Bio SE would yield better results since they have relevant questions (like this one).
What you seem to be talking about are using path lengths in phylogenetic trees to estimate evolutionary distances. However, these trees tend to be constructed starting from some method of computing distance (e.g., genetic, see the Fitch-Margoliash method). Clote and Backofen's Computational Molecular Biology has a brief overview of the area.
In Distance Methods for Inferring Phylogenies: A Justification by Felsenstein, some discussion is given to the exact meaning of path lengths in reconstructed phylogenetic trees. You can, for example, use the path lengths for evolutionary tree dating. Maybe looking into "phylogenetic path analysis" will turn up more work relevant to your interest.
As an aside, there is also work on computing distances between phylogenetic trees themselves; e.g., via the Robinson-Foulds metric.