Some yes, some no.
I.e. Applications developed by RIM (Facebook, live messenger, blackberry maps etc) will require a BB data plan to download and run.
Other applications may not require a BB data plan to run, but must be configured within the application.
Other applications just cant run on wi-fi.
Get the Blackberry data plan, and problem sorted.
"That's it then. Cancel the kitchen scraps for lepers and orphans, no more merciful beheadings... and call off Christmas."