Resolution:
Yes, you have to restart the InConcert Server when the Database server is restarted if the version of InConcert is less then 7.1
Also make sure that you restart the InConcert server only when all the InConcert processes (eg. worker.exe, dispatc.exe and eathread.exe for Windows) are down.

Note, the InConcert server should be taken down BEFORE the database server is taken down.  This will avoid the creation of a large status.log.  The InConcert server will not lose any data nor will the database become corrupt if the database server is taken down before the InConcert server.

With TIBCO InConcert 7.1 the server will automatically try and reconnect to the database when any API request is received.