Gets too much data

I think you'd want to use parens around the (and or) so it will use one OR the other, but not both:
SELECT* FROM ( `morning_qa` )
WHERE`qa_date`>='2014-04-24'
AND`qa_date`<='2015-04-24'
AND (`qa_type`='iris' OR `qa_type`='cones')
ORDERBY`qa_date`;