The sysadmins here were doing some software upgrades the other day and asked me which versions of Tomcat and Apache Solr they should install on our shared Drupal hosting servers. The answer is a little complex, so I finally just drew them a diagram.
Note that in Drupal 7 there are two completely unrelated modules which can talk to Apache Solr: Apache Solr Search Integration, which has existed since Drupal 6 and is primarily developed by Acquia, and Search API Solr search, which uses Drupal 7's Search API family of modules. Both have their advantages and disadvantages, but overall the Search API approach provides more flexibility because it provides a standardized way for module authors to customize the results from several different search backends. With Search API you can also switch search engines with relative ease, should you ever choose to do so.
Here's the chart (click for a larger, more readable version):
In short, any version of Tomcat is fine (but read this note if you're using Tomcat 7). Apache Solr 1.4.x can be used with any of the Drupal modules which provide Solr integration, but only the Drupal 7 modules support Apache Solr 3.x, and Search API Solr search currently requires this patch if you want to do that. Displaying search results using Views is easy in Drupal 7 but requires lots of unsupported, deprecated modules in Drupal 6. And don't forget to update your schema.xml and solrconfig.xml files on your Tomcat server when you upgrade your Drupal Solr modules or switch between Solr 1.4.x and Solr 3.x!
I didn't include any notes on indexing attached files using Tika, but when running Tika as a stand-alone Java executable on the web server you should be able to use any version you like. I had trouble getting Tika to run on the Tomcat server side with Tomcat 5.5 or 6 using Solr 1.4, but a co-worker has a recent version running fine with Tomcat 7 and Solr 3.x.