There are others who can probably answer this better than I can, but my understanding is that the point of the pipe is not to minimize oxidation by isolating your workpiece from the atmosphere in general. Rather, the idea is to isolate the workpiece from any oxidizing conditions within the forge.

In a gas forge, if the burner isn't properly tuned, you have either too much fuel or too much air. If you have too much fuel for the available oxygen, you get a reducing atmosphere and too much carbon monoxide. If you have too much air, you get an oxidizing atmosphere, with all the fuel consumed and an excess of superheated oxygen. That oxygen reacts with the heated surface of the metal and oxidizes the surface.

Now, please remember that in a gas forge, the workpiece is heated partly by the energy of the flame itself (convection) but mostly by the energy coming off the heated refractory inside the forge (radiation). Adding a pipe takes the convection out of the equation, as the only heat getting to the workpiece is whatever energy is absorbed by the outside of the pipe and radiated into the inside. It's true that there's going to be some amount of heated air within the pipe, but it's not going to be anywhere near the volume of superheated oxygen that an oxidizing flame pumps out. Even that small amount can be lessened by adding some charcoal to create a reducing atmosphere within the pipe.

However, because exposure to an oxidizing flame isn't an issue within the pipe, closing it off isn't going to make any real difference in how much the workpiece oxidizes (especially since you're going to be grinding the surface off anyway). In fact, you'll be creating real problems for yourself, since it will be a LOT harder to quickly extract your workpiece from the pipe and get it into the quench as fast as you should.

In short, keep the outer end of the pipe open so you can put the workpiece in and take it out easily, and stick the thermocouple in the open end. If you're doing anything that requires a fancier setup than that, you really should be investing in an electronically controlled furnace anyway.

To pile on a little, no matter how well the flame is adjusted, oxygen and fuel are both present anywhere there's active combustion.  And to an oxygen molecule, your workpiece looks like fuel.  At least one thing the pipe does is lengthen the gas path between the burner and your workpiece, helping ensure it's exposed to the products of combustion rather than the ingredients for it.