Do you have to upgrade to Joomla! 1.5?

From Joomla! Documentation

The short answer is "no". If you have an existing site running version 1.0, you do not need to upgrade. Version 1.0 will continue to be supported for bugs and security issues for a reasonable time. However, all new development is being done for version 1.5 and, soon, version 1.6. Also, most development work on new Extensions is being done for 1.5.

So, if you have an existing site and don't plan any changes to it, you don't need to upgrade. If you are starting a new site, or if you have an existing site but want new functionality, then you should upgrade.

The Joomla! Bug Squad will add an FAQ page for each release to collect issues and proposed solutions that arise. If you are having problems with a current release, check that page for possible solutions, or add the issue you are finding. Continue to post requests for assistance in the forums.

More information about version 1.5 can be found in the Version 1.5 FAQ pages.