I recently replaced a house trap, and as a result, I have a leftover vent pipe that runs from my basement up through my roof. Would it be possible for me to somehow connect a coax cable to the pipe so that it acts as an over-the-air TV antenna? If relevant, I believe the pipe is cast iron but not 100% sure on that.
There is a LOT that goes into making an antenna work properly. That's why the old-school TV antenna has so many little rods sticking out the sides - they have to be certain fractions of the wavelength you're trying to receive or they just won't work properly. Trying to use any old piece of pipe at whatever random length it happens to be will only lead to more tears and frustration. If you want to find out more, go ask over at Amateur Radio, I'm sure they've got a ton of questions about antennas and people who would love nothing more than to expound on it.
Mount a proper outdoor TV antenna to the conduit, run the coax cable into the house through the pipe. Seal the top of the vent to keep rain & critters out.
If you want to try this and see what happens, all you need to do is take the end of your coax and touch the center part of the connector to the pipe, while making sure the outer sleeve of the connector doesn't also touch the pipe. Moving the contact point up and down the pipe will change how it behaves.
That said, there are two reasons to expect this not to work very well:
A vent pipe is quite long, compared to the wavelength of the signals you're trying to receive. Longer is not better for antennas — when an antenna is several times the wavelength of the signal, it will have a fairly unpredictable “gain” or “antenna pattern”, being very inconsistent about what it receives well.
(But for closely related reasons, the idea mentioned above that “the vent pipe is grounded through your plumbing system - basically shorted to ground” is not true — a ground connection multiple wavelengths away will have some effect, again practically unpredictable, but not necessarily a “you receive no signal” effect.)
The pipe is vertical, which makes a vertically polarized antenna. Polarization is essentially a sort of angle the signal exists at, and TV signals are by convention transmitted with horizontal polarization. That's why all the outdoor TV antennas you see have “antenna elements” (rods or bowties) sticking out horizontally. (If you've ever noticed that sunglasses can make your computer or phone screen unreadable — that's the same thing, polarization of photons, and the same sort of problem.)