We just started messing around with this today, and got the same results.
I did however manage to perform a successful tweet when logging in to the advanced mail client with the admin/root account.
From this:
Zimlets Proxy Servlet - Zimbra :: Wiki I suspect a misconfiguration surrounding the allowed domains list, but am sceptical as it worked for the root user.
Working with Zimlets - Zimbra :: Wiki
States that:
Note: If you have more than one custom Zimlet, you should rename the config_template.xml file before updating the configuration in LDAP so that files are not overwritten.
Which worries me somewhat, as the configuration for "allowedDomains" for the proxyServlet may be getting overwritten depending on the order my zimlets are being installed. I am unsure of how it is stored in ldap, however, I would rather assume that each zimlet has its own ldap entry based on its name, eg: <zimletConfig name="com_vtwitter_vtwitter" version="0.5">
Then the above statement must be false, and not of any consequence.