I don't think the parks really do much to celebrate a new movie release in the parks on the day of the release. There is already a fair amount of merchandise out, they have a caramel apple for both Dory and Nemo, and they're doing a special "Speak Like a Whale" day this upcoming Saturday, which sounds kind of cute.
I was at Disneyland today for the advance screening of Finding Dory and as you entered into the parks they handled out buttons celebrating "Speak Like a Whale Day, June 11." I'm not sure if they will have anything special on the actual movie release day though.