Can non-teaching staff write teaching recommendation letters for academic jobs?
I would view the letter favorably if the letter writer made it clear that they had observed the candidate's teaching.  

I've seen some TA training programs where the TA's are observed in the classroom and get formal evaluations by someone involved in the training program.  A letter of recommendation from such a person would be quite valuable from my point of view (and assuming that the letter was positive this would be helpful to the candidate.)  

On the other hand, if the person that you're considering is just a faculty development specialist who runs workshops that you've attended and hasn't directly observed your teaching, then such a letter would be much less useful (read useless) to me.