Some of this question would be highly application-specific, but what approaches do you take when looking to migrate applications from one server/platform to another and servers form one network segment to another?

For applications that can't be re-IP'd (many exist in this category), the general answer is to nuke and pave (or extend a clusterable application, then remove the segment that needs to be "moved").

For "normal" applications (httpd, mail, directory services, etc), what are the checks ou perform before, during, and after a move to ensure the health of the migrated app/server?

An example with Apache:

1. backup httpd conf directory
2. change httpd conf files to use new IP address of server
3. change (or add) IP of server
4. restart Apache
5. verify web server still serves pages
6. reboot server
7. verify environment comes back up healthy