What would cause the automatic condensate pump to all of a sudden make a tremendous amount of noise? Is it safe to shut the furnace off and unplug the pump?

Probably the condensate pump went south.

If the furnace is gas and there is no humidifier, you should b OK turning off just the pump. If there is a humidifier and a gas furnace, then turning off the pump should be fine too.

It also depends on where it's located and if there is a pan switch that prevents flooding.

To answer your original Q, you can turn off the furnace (thermostat would work) and unplug the pump.

Heat pumps and humidifiers generate water. Those should be turned off as well. If you have a HP switch to emergency heat and turn off any humidifier.