If you are dragging an application or file icon to the Dock, the normal behavior is for it to stay in the Dock, even after a reboot.
If you are just opening an application or file, then an icon will show up in the Dock, however, it is only temporary. When the computer is restarted, the applications close, and the icon will disappear from the Dock.
Unless there is more information to the puzzle that you have not presented?
Dragging and dropping an App to DOCK should stick, unless the DOCK plist is screwed up.
Go to Folder from the Finder's Go menu > Enter "~/Library/Preferences/" without the double quotes and enter. When in Preferences folder move the com.apple.dock.plist file to the desktop. Open terminal and enter the command "KillAll Dock" without the double quotes and try dragging and dropping the app and test after a reboot. (sometimes you may need to restart the computer after moving the plist from the preferences folder)