I installed JMS 1.2.33 in Joomla 1.6 Beta5 and it shows a successful installation.

Then when I go to Administration | Components | Multi Sites and click on any menu entry I get a popup window asking me to download 'index.php'. It's like there is some type of mime error.

Trying to download yields a 0 byte file.

The only error in the error_log is:
[Sat Jul 31 17:29:14 2010] [apc-error] Cannot redeclare class jcachecontroller

Not sure what that error is but the rest of my sites all seem to be working without any problem.

On the chance that something unnoticed went wrong during installation, I uninstalled JMS and reinstalled it. But still have the exact same problem. It didn't make any difference.

Install the JMS 1.2.34 that should work better.
The 1.2.33 were working on Joomla 1.6.0 beta 3 but not on the other versiions.

I am currently preparing a JMS 1.2.35 that will solve other problem detected with Joomla 1.6.0 beta 6

See change history
www.jms2win.com/faq/change-history-v12x

See also the procedure to get the latest version
www.jms2win.com/faq#sec-122

I went to the 'get latest version' link and clicked on Get Latest Version button but it still shows 1.2.33 as latest version. How can I get the 1.2.34 version to show up?

You have forgotten to click in the check box.
When you do not click on the product for which you want to get the latest version, it resend you the same version that you previously downloaded.

Checking the checkbox fixed the update issue.

However, uninstalling 1.2.33 and installing 1.2.34 did not fix the problem.

All of the links on the Components Menu work properly except for MultiSites. The MultiSites entries all try to download index.php as a 0 byte file.