Can a formula be used to check the status of a checkbox in a long text field?

Kind of.  The ticked checkboxes are technically "[x]", so you could use a formula to check for that.  Could you provide some example data as well as what you would like to happen?

Thanks for replying. I have individual patients with corresponding long text fields containing a checklist of action items. Once an action item is checked as completed, I want to have it move to a 'historical action items' column. 

I have created a formula checking for checked boxes, but this results in all the the items moving once only a single one is checked. Is it possible to specify to look at checked boxes and ONLY move those items while leaving unchecked items in the original column?

Hmm....yeah this is doable.  I would:
1. Use a formula field with REGEX_EXTRACT() to grab the line where the checkbox was marked
2. Use a formula field to display the same checklist without that extracted line with a SUBSTITUTE()

Then I'd use an automation to:
1. Add that value from the formula field from step 1 above to the "Historical Action Items"
2. Update the original checklist field with that updated checklist from step 2