I'm breaking your question into the two major parts within.
"When testing an amateur radio HF or VHF antenna, how would one
absolutely minimize the amount of any RF radiation from the feedline..."
- I agree with Mike Waters about the judicious use of proper ferrites near the feedpoint and every 1/4 wave down the feedline to help quell any induction between feedline and antenna under test.
- Perform some sort of EiRP test using as small a transmitter as possible to simply eliminate any feedlines. Perhaps a small HT for VHF and a Tiny Tin assembly for HF. Yeah you still need a way to receive the power, but if you compare the test antenna with a reference dipole or equiv. the results are often quite revealing.
Parasitic conductors and dielectrics
"...tower, and/or any other parasitic elements, such as power supply cords, AC
wiring, computer cables, gutters, or any other nearby antennas?"
The goal here is to detune anything that might resonate at the frequency of interest. Ferrites often help with this. Sometimes the lengths are already non-resonate to mess with your goals and you might actually cause a resonance with a ferrite. Best if possible to remove these objects from the near field of the antenna or just test with them as part of your overall antenna "system."
It's what we do at the antenna lab
Eliminating the feedline is job number one when we perform testing at work. The EiRP test, such as this one testing HT Antennas, is our favorite way to really see how things work without benefit of additional conductors. We get our fair share of snake oil antennas that completely rely on feedline radiation to work. Our test snags them and quick.