Section 1
The first section of Stage 14 is unique, at least in the Japanese version, in that it is the only section that requires you to find a Sphinx in order to escape the section. In the American version, this aspect was removed, and the door is immediately open. In the Japanese version, you'll have to search the block underneath the very first treasure chest that you see (which contains a 10,000 point balloon) in order to find the Sphinx that will allow you to continue on to the next section.
 Section 2
Not so much a secret, but rather a useful tool, the right end of this branch can be removed to provide another exit from the otherwise one-way subsection at the bottom. You would be wise to create this extra exit before you venture into the space below so that you don't end up trapped by a swarm of enemies.
To access the slavery room below Section 2, you must do one of two things. If you are playing the Japanese version, you must be touching the door when the Timer reaches 60. Since you're unlikely to arrive before 60 comes and goes, you may need to collect some of the Mighty Drinks stored in the section to boost your Timer above 60 again. In the American version, you must find an unfortunately positioned Sphinx. It happens to be in the block immediately to the right of the exit. In order to get to the block without triggering the stage exit, you must start pushing right as soon as you are lined up with the block. Otherwise, you will be sent to the bomb room. If you push right at the correct time, you should land on top of the block instead, where you can jump and reveal the Sphinx, and head back to the slavery room.