Why does fluid under pressure not flow into closed-off tube? A set-up is shown:

I plugged the end of the tube, and slowly started pouring in the liquid. However, as I poured, no liquid entered into the chamber of the tube even with the pressure acting on it. The fluid is viscous and the tube is relatively small in radius.
Why does this happen? My guess is that the pressure in the tube ended up being greater than the static fluid pressure. Could it be because of surface tension?
 A: It is indeed an issue of surface tension.  In the setup you describe, no amount of pressure will force the air out.
As you add fluid, you compress the air evenly.  The air will get compressed into the tube until the pressure of the air is equal to the pressure of the liquid.  From this point of view, there is no reason for the air to ever leave the tube.
What you really end up needing to look at is the differential pressure between the top of the tube and the bottom.  This is, of coursed, fixed based on the density of the liquid.  As you add more liquid on top, it causes both the pressure near the top and pressure near the bottom to rise equally.  The difference is always $\rho g\Delta h$, where $\rho$ is the density of the liquid, $g$ is the force of gravity, and $\Delta h$ is the difference in height between the top and bottom of the tube.
This is obviously a very low pressure, because the tube does not have a very large diameter.  This is where surface tension starts to play a part.  In order for the air and liquid to change places, it must go through a transition where there's far more liquid-air interface.  If the surface tension of the liquid is large, this will prevent the bubble of air from rising to the top of the tube and sneaking out while the liquid fills in from below.
One solution I used recently to this is to put a smaller, longer, flexible tube into the tube you drew (a straw can work, if your tubes are large enough).  You have one end of this tube at the back-top part of the tube you drew, and the other end is in the open air above where you put the liquid in.  When you fill the container, this gives the air a way to sneak out which isn't blocked by the surface tension of the liquid.
This effect is really easy to notice if you fill a pair of two-liter bottles with water and then stick a long straw into one of them.  Turn them both upside down to empty and the one with the straw empties much faster!
