Why/ why not? Since I am the only Staff on mine, I do read all the posts. I have a system so it doesn't feel so overwhelming. If I ever got bigger & had Mods to help out, I may still read all posts just to make sure it's being moderated well & no spam is slipping in.
I always strive to read everything on my forum, even if sometimes it may take me some time to get through everything if there's a lot of stuff to catch up on and my time is limited. Once in a while I might miss something entirely, but I try to do my best to check everything out and respond if I have something to add as well.
If I were actually running my own personal forums right now and actively attempting to advertise, I'd read as many of the topics I can, if nothing gets in the way. At a certain forum owned by a banned member, I learned that if I wanted to read everything posted there, essentially I would have no time to attend to even my basic needs, to stay on top of the stuff posted there, when it was at it's most active point.
I read all posts on my forum, and if required I even edit the posts. That's because I don't want to give an impression to the visitors that the forum does not have useful content. I want to rank every thread on the search engine so that I can generate organic traffic.
I kinda do, as long as I'm alone in the admin chair. As the activity will pick up speed and we'll have a mod team in place, I can relax a little more. I do love reading forum posts though, so it's never a chore.
I always read all the posts on a forum that I am staff on mainly because I want to be sure that nothing falls through that shouldn't be on the forum. On forums I am not staff on I will read through posts I am interested in and that I intend to reply to but I don't generally go round and check every post unless I have an interest in it.