Not that this matters for you, and since this is about 1+ months old, I wanted to post here so future viewers are aware as well.
We had the same issue. Initial tweet would go out, but any future tweet would not. After a really long investigation last night it was actually two factors that played into it.
Factor #1
WP_Cron – We were using a system level cron job, following a tutorial from here: https://developer.www.ads-software.com/plugins/cron/hooking-into-the-system-task-scheduler/
However, the issue was with our host now allowing our tasks to fire from crontab under our user. Updating the code set the user as root for the cron fixed the issue for us:
*/5 * * * * root wget https://mysite.com/wp-cron.php &> /dev/null
With a quick cron testing plugin, I found the cron to be firing at this point.
Factor #2
W3TC – There’s a known issue with W3TC and cron jobs which use object data. WordTwit is one of those issues. See here: https://www.ads-software.com/support/topic/self-diagnosed-and-fixed-w3-total-cache-bug-in-faulty-object-caching?replies=9
Once I disabled database & object cache, our tweets started going out automatically again.
I elected to edit W3TC myself and put in the first patch mentioned by archon in tha thread. Once I did that, and re-enabled object cache, things seem to flow normal again ( with the added system cron fix from factor #1 )
