First, consider the following:
1) Is it possible that the controls have been changed so that another key would trigger this function? If you select any other slot with an item in it, can you place it by right-clicking?
2) Are you running a text-based mod that requires the chat area? Can you see the chat area? (Try typing / then enter ) Is it possible the game settings have been changed so that the chat area is not shown?
2) Try disconnecting from he server and then reconnect, just to be sure.
If the game's settings have been changed then the solution is to uninstall Minecraft then redownload and reinstall it from minecraft.net.
Users with paid Mojang accounts can download it free: select 'Already bought the game? Download it here' from the minecraft.net homepage.
If those things doesn't resolve your issue, then:
1) Try typing in the command /clear. Then go back to LearnToMod and press the orange Mod if another mod chest does not appear.
2) When logged into Minecraft, press the Esc key and go to Options, then to Controls. Reset the control for "Use Item/Place Block" so that it says Button 2.