I'm building a company website aimed at consumers. I plan to use PHP
or ASP. How do search
engines deal with ASP and PHP pages, compared to static HTML? What if
there is dynamic data that changes periodically - will that affect how
the site is listed?

curiousdave-ga,

The search engines are looking at several factors, not just the page's
code generation, namely the number and type of other links to the
site, the meta tag (HTML) keywords (even if the page is PHP/ASP
powered, it still generates HTML), site content keywords, page titles,
and possibly page file names.

omniscientbeing-ga

Being a .php or .asp page will not making any difference, but being
dynamically generated might. A fundamental principle of search engines
is to present results that reflect what is there now, not what was or
what was due to specific criteria. Google is known to have an aversion
to indexing dynamic pages that have "ID" somewhere in the URL.

If possible have the main pages of your site static, and the product
pages dynamic, and get the product pages indexed by Froogle.