Of course, the answer to this question is certainly "it depends."
Yes, different antenna elements in close proximity can interact and cause interference problems. The most notable issue would be a degradation of both antennas. You don't want that.
But you might not have a problem - again depending on what sort of antenna you wanted to put up, and what sort of orientation the respective antenna elements would have with one another.
Now, I'm going to assume that you are not using the TV antenna at all, so let's not worry about any deleterious affects there. The primary question would be, would the TV antenna affect an antenna hung from the same support.
For example, if you used the TV mast to anchor one end of a 40m dipole, I doubt you would have any problems, particularly if the dipole element tips were 5-10 feet away from the TV antenna.
If you used the TV mast to anchor the center of a 40m Inverted-V, you might have a little bit of interaction with the TV antenna -- moreso than you would have with the ends of the 40m dipole.
That said, we Amateurs typically mount several antennas on a support, generally with some vertical separation. At my station in Gwinnett county, I have a 15m (50 foot) tower with a Cushcraft A3S / A743, an 80/40/20m trap dipole with one end anchored at the 12m (40 foot) level of the tower, and a 30/17/12m inverted V, whose apex is at 13m (43 foot) level of the tower. All of these antennas are in close proximity, but are mostly for different bands (the dipole and Cushcraft have 40 and 20m in common). From my testing, I don't see any interaction between them.
If you aren't using the TV antenna, I would suggest you take it down and use the mast as a support for whatever antenna you like.