Tool to manage and/or make available a list of my publications on the web?

You could consider hosting it via one of the reference / citation managers, such as CiteULike or Mendeley, which can take imports of Bibtex files. Your own university may (should!) have such a web-front plus publication database available, for you to embed into your university home page.

Wordpress

If your webpage is hosted directly on wordpress.com, then no, you can't host your own PHP script. If it's Wordpress hosted elsewhere, then you should be able to add your own PHP script, by incorporating into your plugin. There are skeleton plugins available to get you started. It may be that asking on Wordpress.SE will give you some useful pointers.

Mendeley

On Mendeley: there is an embedding plugin for Wordpress. I haven't used it, but it might be worth looking into. Or, on your Mendeley profile web page, select edit, then embed. Or Share > Embed elsewhere on Mendeley pages (groups or whatever).

There's an article on Beta Science on embedding Mendeley that may be useful. It uses the same Share > Embed as above, but recommends creating a "Publications" group in the desktop client first, putting your own papers into that, (in order, from oldest to newest). Then right-click -> Edit Settings, then under "Collection Access" choose "Public - visible to everyone". Then click "Apply and Sync". Then, from the collection web page, select "embed" to get the appropriate html.

CiteULike

If you prefer CiteULike, they have an excellent API, allowing you to customise your own tools. And the staff on the discussion forum are very responsive (a marked contrast with Mendeley).

Academia.edu

If you prefer Academia.edu, you could always embed your publications page (and maybe your talks page too) in an iframe

You can also try out Google Scholar Citations, which is a new service for showing the citation count for all your papers. As an example, here's my small list.

Bibbase

I think that BibBase would be a perfect solution if you can at least run CGI scripts; instructions can be found on this page (if you can run PHP then it's even simpler). You need to feed it a Bibtex file, but if you use Mendeley then it can automatically grab it from there.

Update: Bibbase now allows you to use just javascript. (h/t ChristianF)

BibBase makes it easy for scientists to maintain their publications pages. As a scientist, you simply maintain a BibTeX-file of your publications, including links to the papers, and BibBase does the rest. When a web user visits your publications page, BibBase dynamically generates an always up-to-date HTML page from the BibTeX file, and even allows the user to sort the publications by other than the default ordering (e.g. year, author, keywords, research area, publication type).

Here is an example of the output.

A custom alternative

I happen to have a departmental server that doesn't allow PHP or CGI; you can read about what I do to solve this problem on my blog and see the kind of output generated on my group's site. An even more jazzed-up version, which is searchable and filterable, is on my own site.