Why does third gear in your 67 Mustang make a whining noise?

Could be one or all of these items.

Oil level too low- you'll notice 3rd is much higher than 2 and takes you right up to 50 mph or more, this increased speed in the gears can cause starvation of oil to the lower end of the gears.

Bearings worn or damaged. How old are they?

Gear teeth worn or damaged. again, they should be quite old at this point.