It appears that there is not an adjustable setting that controls whether or not the open documents are remembered. By default, Visual Studio stores this data and then re-opens the documents that were open the last time the project/solution was closed. As pointed out by some links referenced by Stuart Dunkeld, it takes a Macro in Visual Studio to override this default behavior.
My problem appears to be that my 'Settings' had become bloated/corrupt. Based on the suggestion of Luke, I exported my settings, then did a reset. This restored the default 'Remember Open Documents' behavior. Then after much fooling around, I was able to import (almost) all of my old settings and still not mess up the proper document behavior.
When importing my settings, I selected everything except the 'Window Layouts' settings.

This restored my existing settings, while keeping the default Visual Studio behavior of remembering which documents were open.
I then re-exported my (fixed) settings so that I had a backup. That's when I noticed that my old, possibly corrupt exported settings file was 8.6MB, while my new, correct exported settings file was only 0.2MB.
After examining some of the differences between these two settings files, some of the information leads me to believe that the installation of VS2010 SP1 had a conflict with the 'Document Well 2010 Plus' feature in the Productivity Power Tools extension, but I cannot positively verify this.