Melissa,

As you know spring and summer are much better for blooming flowers. Although when I visit Irish gardens in fall there are certain things still blooming and berries coming to fruit. I suppose it very much depends on how the summer went. But the Burren has a rather upside down environment and farmers put animals out to pasture on the hills in winter. So things are growing but maybe not the flowers you saw in June. 

Try these websites for some insight on the Burren:

http://www.burrenbeo.com/

http://www.theburrencentre.ie/

http://www.burrenwalks.com/

Maybe if you email one of them you can get some current info about whats going on in the Burren. 

Michele