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Not in my experience, not unless it is BOTH old-style p/t (CCA) AND extra-heavily-treated for burial.
Honestly, all concrete really does is provide a bit of extra weight vs *lifting* forces, which a run does not normally experience, and make it a heck of a lot harder to reset, repair, replace or remove the post. (And it's an extra expense). In the case of a deck or pole-built building or some such thing, a concrete footer (*under* the post bottom) also helps spread the load to keep the posts from settling unevenly, but again, not really an issue for a run.
If a person has oodles of cement mix sitting around and is either convinced they will never ever want to remove the posts and just abandon them once they get broken or rotty, or is looking for some good exercise <g>, then I would not stand in their way of setting posts in concrete, of course.
I just don't think it really serves a useful purpose for a run fence, and has some undeniable disadvantages.
JMHO,
Pat