How can I tell whether a DC device will accept a greater voltage than stated?
The simple answer is you cannot know without understanding the circuit involved. 

For instance, your scooter controller probably uses PWM to control motor speed, and the FETs used for power drive will almost certainly handle 36 volts as well as they do 24 - the lowest-voltage parts made usually start at about 50 volts, and it would cost more to get special parts than to use these over-rated parts. The same probably goes for the lower-level parts. 

But maybe not. 

Even if the controller parts will handle the increased voltage, it's entirely possible that the wiring won't, where "won't" is synonymous with "will melt the insulation and catch fire". And the motors themselves may well be damaged by application of over-voltage and/or over-current. 

So the answer is really "no" on several levels. Not only can you not tell by simple, ignorant inspection, there are parts of the system other than the controller which may be damaged even if the controller survives.

You should take @Brian Drummond's phrase "smoldering remains" very seriously.