Why do people remove the horizontal output tube before powering on a vintage TV using a variac and dim bulb tester?

Because the horizontal drive needs to be normal for the horizontal output to operate normally (full on/full off) so that it won't overheat.

Should the vertical output tube also be removed while using a variac?

Some people also remove the vertical output tube...

You can't effectively do this on series heater sets without jumping heater leads back into circuit.

The set ironically where pulling the horizontal output matters most is the CT-100 that was designed so that even if the osc was running if the line voltage was around 90V the bias would be badly wrong on the output.

My approach depends on the value of the set. Something rare and desirable is far more likely to get this treatment than say a run of the mill monochrome set that's common and cheap.