I run a Minecraft server on my home computer and if I make a change to the whitelist from inside the Minecraft client I must reload it for the change to take place. The Minecraft Wiki states the command "whitelist reload" is used if changes have been made to the whitelist.txt from outside the Minecraft client or server.
For example a friend a school asked to be whitelisted and I did from inside the game. He tried to connect and the server kicked him for not being whitelisted. I restarted the server and then he could get on.
This is another story inside the Minecraft server console. You don't need to reload the whitelist if you modify the whitelist from the console.
Steps to reproduce:
1. Make a server and whitelist it.
2. Add a person to the whitelist from inside the Minecraft client.
3. Have the person try to connect.