Should the database service be restarted after performing a restore?
There is absolutely no reason to.

Once the database is restored, if over top a previous database or as a new one, there will be no information in cache (or very little). If you were to restart the services, the only thing accomplished is that SQL Server has to run recovery on all of the databases in the instance and re-acquire memory. This would actually cause a detrimental effect, assuming this isn't express edition and the buffer pool is actually being used (i.e. there is more than 1 databases on the system). Even if it were, there is still no good reason to do so.