I know you can use a medic to Blind a unit, but does anyone know of a way in NON-Brood War to reduce a unit's sight range, other than taking advantage of upgrade differences?

Other than reducing a unit with upgraded sight to non-upgraded sight, no, or you could reduce them to burrowed unit sight, since that is less than normal, but nothing that is as small as blindness.

I see. That's what I thought. Unfortunate...

Is it possible to reduce a unit to burrowed unit sight without actually burrowing it? Or to do it to a unit that's unburrowable, like a zealot? I need reduced sight on a unit that the player can move around.

Have a burrowed unit follow your unit, turn off vision with youself using triggers, and turn on vision with the burrowed units player

Mines have pretty small sight range

That's one way, another is to simply create the burrowed unit for a player that has burrow researched, then give it to the player with burrow disabled, now they have a burrowed unit and they can't unburrow.