Probably not what you are looking for, but:

There is this recent paper by George Raptis:

<a href="https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/Homotopy-theory-of-posets/hha/1296223882.full">Homotopy theory of posets</a>, 2010

discussing about model category structures on the category of posets.