Burn smell after cleaning wiper blades and windshield?

Sounds about right, the smell would dissipate quick because there isn't a lot of mold coming out. The only time I've seen mold in a washer tank was when people ran some of the oddball washer fluids that were scented. The common stuff you get is water, alcohol and dye. Some of the other ones that deice or contain something like rain-x could have other ingredients.
There is no rubber in or around the motor that can overheat, the only rubber like item is inside the pump itself (the impeller) and it is surrounded by washer fluid so there is no way it can get hot. Generally the washer pumps are not connected to the wiper pumps any longer. Instead the wiper motor is smaller and lives under the cowl. The washer pump(s) are mounted right to the washer tanks, usually buried behind the fenders. With those you wouldn't smell anything, they just fail because of the tiny wire used to wind the motor. So you go to use the washer and they just don't work. That commonly happens in my area when folks fill the washer bottle with "southern" fluid IE the stuff that isn't rated for extreme cold. Then it either turns to slush or freezes solid. You hit the switch for the washer and the instant high load of the locked pump causes the winding to act like a fuse. Even then there isn't any burning rubber smell, it would be an instant tiny pop and you might smell the acrid smoke if you had the pump in your hand. Once you smell it you would never forget it. Between working on vehicles, welding, painting and being a firefighter-EMT for 20 years my nose can tell me a lot about whatever I smell, unfortunately some of those scents also haunt you because of what they were caused by.

No it is much different than pretty much any other odor. More like a sharp burnt plastic with heavy ozone mixed in. Not easy to describe, but once you smell it you will know.