It does not seem as if "posets" and "order-theory" are entirely synonymous since the tag order-theory seems to encompass more questions than the tag posets. These tags should be kept as they are.

For example, a question about Heyting algebras, frames, or Boolean algebras could justifiably be tagged as order-theory. On the other hand, for several reasons, it is harder to see questions about Heyting algebras, frames, or Boolean algebras tagged as posets since Heyting algebras, frames, and Boolean algebras are very specific types of posets. Furthermore, structures such as Heyting algebras, lattices, and Boolean algebras can and usually are defined in terms of their lattice operations instead of their partial ordering. Therefore for these structures, the tag "order-theory" will probably be more appropriate than the tag "posets".

I should also mention that there are structures similar to partial orders which are not quite partial orders. For example, as mentioned before, preorders, and cyclic orders fall under "order-theory" but these objects are not quite partial orders. Also, sometimes partial orders are endowed with extra structure and these objects should not be seen as simply partial orders (i.e. partially ordered topological spaces).

Introduction to Lattices and Order. People in different areas use the word "order" in slightly different ways (just as people in different areas use the word "ring" in slightly different ways, but "ring theory" is still a useful term). Presumably most mean partial order, but some might mean total order, preorder, etc.; they're all bound up together. $\endgroup$