This is strictly recommended for advanced Mac users. In general, if the app still throws such error messages, it's better not to use it or use it at your own risk.
With the command line you can use xattr to view and remove extended attributes from a file on the Mac including the application throwing the “Appname.app is damaged and can’t be opened. You should move it to the Trash.” error message.
Launch Terminal and then type the following command:
xattr -cr /path/to/application.app
xattr -cr /Applications/FileZilla.app
The -c flag removes all attributes, whereas -r applies recursively for the entire targeted .app directory contents.
This is only recommended to advanced users because modifying extended attributes may have unintended consequences, and again you might be attempting to run an app that you should not be running, either for stability, privacy, security, or other reasons.