Effects of making a hole in a mercury barometer
The simple force diagram that we rely on to prove that we can measure air pressure with this device gets far more complicated when you start drilling holes.
Consider doing this with water. Water behaves reasonably intuitively. If you drill a hole in the end of a water barometer like this, what you expect will happen. Water will pour out of the hole, and then all the water will drain from the tube.
The reason for this has less to do with the top and bottom of the barometer and far more to do with the top and bottom of the hole. The pressure differences there are sufficient to permit a water/air exchange, letting air into the tube.
Mercury, however, has an extraordinarily high surface tension. It loves to attach to itself and cares much less to interact with air or the edges of the tube (for most tubing materials).
Now our model is more complex. What you will find is that the forces on the mercury near the hole are affected not just by air pressure but by the surface tension as well. The mercury will try to keep the surface area of the mercury/air interface as small as possible, preventing air from going in or mercury from leaking out.
The result? The barometer will probably remain intact. The surface tension effects will provide the additional forces needed which the air pressure does not, and you'll see no significant change.
This, of course, depends on the size of the hole. Make the hole too big, and the surface tension will no longer be able to hold the mercury in.
By the way, mercury spills are bad news. I highly recommend not drilling into mercury barometers. Its really nasty stuff and needs to be disposed of properly.