Eventual consistency in databases is fully described by "make your merge operation the join of a semilattice." I once watched a whole keynote at Strange Loop which slowly recapitulated that idea over the course of an hour when with lattice theory it's just one obvious sentence.

I haven't written it up, but I used lattices as the starting point for an algebra of genome annotations that I personally found useful. I had to replace one of the two operations with a different one, but guiding it as close to a lattice as possible was a useful heuristic.

The tree of life is usually described as a tree of species. Defining a species is a problem, though, especially for microbes, and there are places where it's not a tree and things hybridize, especially for plants and microbes, so I spent some time trying to define it in terms of individual organisms. I'm still not satisfied with where I had pushed it, but the structure when I had left it took the form of a chain complete partial order.