I wouldn't know why Remote Buddy would ever ask you for an 
administrator password (or any password) on application startup / 
application ending. It will only ask you for admin priviledges when 
you want to install, testdrive or uninstall Remote Buddy's driver and 
works fine even with a non-admin account.

If it is really Remote Buddy that is asking you for an admin password, 
then it must be an indirect consequence of how you set up your system. 
The only thing that could cause this that I currently know of would be 
if you have added an admin's keychain to your non-admin user's 
keychain. In that event, OS X will also always search that other 
keychain for entries and - if it finds one - will automatically prompt 
you for the password for that keychain (which - for a login keychain - 
is always the respective password of the account). The solution here 
is to copy the respective Remote Buddy entries from that other 
keychain to the actual login keychain (via drag and drop in Keychain 
Access.app) of the user that you are logged in as and then remove them 
from the foreign login keychain.

Above, however, is nothing that Remote Buddy has any control over. 
Those are OS X keychain automatisms.

Best regards, 
Felix Schwarz