<p>I'm building a <code>PHP</code> site, but for now the only <code>PHP</code> I'm using is a half-dozen or so includes on certain pages. (I will probably use some database queries eventually.)</p>

<p>Are simple <code>include()</code> statements a concern for speed or scaling, as opposed to static <code>HTML</code>? What kinds of things tend to cause a site to bog down?</p>

Certainly include() is slower than static pages. However, with modern systems you're not likely to see this as a bottleneck for a long time - if ever. The benefits of using includes to keep common parts of your site up to date outweigh the tiny performance hit, in my opinion (having different navigation on one page because you forgot to update it leads to a bad user experience, and thus bad feelings about your site/company/whatever).
Using caching will really not help either - caching code is going to be slower than just an include(). The only time caching will benefit you is if you're doing computationally-intensive calculations (very rare, on web pages), or grabbing data from a database.