You quote from the article, but in the comments the author himself states:
Not using DIVs tends to make sites render more reliably cross-browsers. You’re removing an element in the code that could be the source of a browser not displaying correctly. Remove the variables and you’ll see less problems.
So: switching divs for headings does not change the reliability of rendering (the article implies that, but the author does not mean that), but removing unnecessary nested divs elements help that, but as a good HTML layouter you should always do that ;)
Just keep in mind that you should prevent Divitis whenever possible, and making use of semantically correct markup helps your SEO efforts and accessability and karma and stuff.
EDIT:
OK, as all know, divitis is bad. Let's have a look at the article's markup:
<body>
<div id="page">
<div id="header">
<h1 id=logo">(some stuff)</h1>
<ul id="nav1">navi</ul>
</div>
<div id="columns">
(Content)
</div>
<div class="box6">
<div class="top"></div>
<div class="spacer">
<div class="col3">
</div>
<div class="col3">
</div>
<div class="col3 last">
</div>
<br class="fix">
</div>
<div class="bot"></div>
</div>
<div id="footer">
(Footer Content)
</div>
</div>
(Script tags)
</body>
Let's see: <div id="page">
to center the page with margin: 0 auto;
. Apply this style to <body>
and you can remove one div. The whole content of the <div class="box6">
is not exactly clean of divs, and absolutely unnecessary. And for the rest: See for yourself.