Is my electrolysis emitting compounds from additional reactions? I am separating water into hydrogen and oxygen via electrolysis.

The electrolyte solution is sodium hydroxide (NaOH) in deionized water.

Electrodes are 99.96% pure nickel.

All gases and liquids are perfectly clear.  Even after running for hours, the electrolyte solution is crystal clear and colorless.

However, the reaction seems to be emitting something that burns the nose and throat of everyone in the room.

Is that some sort of gas from a side reaction at one of the electrodes, or is it simply bits of NaOH being carried into the air by the splashing bubbles?
TL;DR: Your suggestion that the problem is "simply bits of $\ce{NaOH}$ being carried into the air by the splashing bubbles" is very likely correct.  

You have a very simple system, meaning a minimal number of variables to consider in order to eliminate other possibilities. For example, if you were using $\ce{NaCl}$ as the electrolyte, then chlorine gas could be the irritant. But with $\ce{NaOH}$ as the electrolyte there really is no side reaction that could produce such an irritant.  

On the other hand, $\ce{NaOH}$ itself can irritate sensitive tissues like mucous membranes, and even skin for that matter. According to one of your comments, you are using a $\ce{NaOH}$ strength of a bit over 1 molar. Droplets of this solution will eventually be neutralized by $\ce{CO2}$ in the air, but at that strength it should stay around long enough to elicit the symptoms you described.