OK, I solved this one myself. Big "derp" moment.
I had updated the version of PHP running on my WHM-managed server. I needed to go back to the Task Method settings inside of my IPB Admin CP and copy the cron task string that it provided, into the definition of the cron job that runs every minute to execute the task in CPanel.
Hopefully this is enough info to help someone else who might run into this issue.
Whiskey Bizness's post in dailyStreamSubscriptions task failing was marked as the answer