I'm using a furnace blower motor that has taps for three speeds and a multi relay board. If (for whatever reason ie bug) I power more than one winding at a time will that damage the motor?
Possibly but without the specs of the motor that's just a guess.
To eliminate that you could use a bank of 3 relays with only one available to be active at any one time and a few ms between changeover.
I should not really have to remind you but the lack of info compels me to do it 
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Sorry don't have the specs but I'll try to find them. Figured furnace speed tapped motors were typically same.
Different windings, different RPM's, different air flows, different voltages (240/110), soft start, hard start, etc.
But regardless of the differences a relay or SSR would still be a good choice at this stage as most other things can be taken care of.
freakdaddy:
I'm using a furnace blower motor that has taps for three speeds and a multi relay board. If (for whatever reason ie bug) I power more than one winding at a time will that damage the motor?
Years ago I salvaged a 4 speed furnace motor from the metal bin at the transfer station. It ran fine and I built a 4 speed cooling fan for my business. I used a rotary switch with an open position between each of the 4 motor wires. Ran fine for about 15 years and was sold at auction when I closed the business.
I suspect the 3-speed motor is similar to the one I found. Your Ohmmeter will let you identify the single common wire and each of the speed connections will each show a similar resistance to the common wire. Try connecting the common wire and each of the speed wires, one at a time, to your mains power, and then you will know which wire gives which speed.
I have it figured out. That isn't the problem. My question is what would happen if you powered two or more of the taps at once. And I don't need to or want to. Does it run slow? Fast? Not at all? Burst into flames? It's just more of a theoretical question.
Theoretically, the fuse you have in the power line would blow.
Paul
That depends on if they are sequential taps or seperate taps with a common ground.
Try think of it as one side of a transformer.
Sequenced
Seperate
And as mentioned fuses will blow (or not)
Without knowing how the windings are arranged the question can't be answered - bursting into flames is entirely possible if simultaneous connection shorts part of a winding, shorted windings get hot very quickly.