<p>I have a simple page with my <code>ScriptManager</code> and my <code>UpdatePanel</code>, and my <code>ContentTemplate</code> has one <code>ListBox</code> and one Label.  I am just trying to catch its <code>OnSelectionChanged</code> so I can update the text in the <code>Label</code>.  No matter what settings I try to tweak, I always get a full-page postback.</p>

<p>Is this really not going to work, or am I just screwing this up?</p>

You need to have Sharepoint 2007 service pack 1 -- or else there's no chance.
(Sharepoint 2007 predates ajax 1.0 -- they built some support into service pack 1)
Next, from a trouble shooting point of view, test that the
exact
same code functions as expected when hosted in a regular asp.net page. (Literally copy and paste your code across into a fresh page, to rule out any typographical errors). Ruling sharepoint in or out of the problem area will help narrow down the solution space.