Does a cake that has strawberries in the batter need to be refrigerated after baking?

No, because the moisture in the (usually pureed) berries is taken into consideration when the recipe is formulated, and baking yields a shelf stable product.

Here are a couple of strawberry cake recipes:
(Using fruit in the batter.)

http://www.recipezaar.com/22984

http://tinyurl.com/2cu8s4

HTH

Thank you so much JanH! I did use the WASC with strawberries added - delicious!!