Can't get my SQL query to filter by dates like I want
!= doesn't support a list of items like that.

Try one of these (note the commas between items):

WHERE Format([Start Date], 'MM/yyyy') NOT IN ('07/2016','08/2016','09/2016',...)


or even better (and faster), without converting to text (if your date range is truly consecutive like your example):

WHERE [START Date] NOT BETWEEN '2016-07-01' AND '2017-07-31 23:59:59'


Since between is inclusive of the endpoints, that is equivalent to:

WHERE [START Date] &gt;= '2016-07-01' 
AND   [START Date] &lt;  '2017-08-01'