What does the value of the pH meter mean when there aren't any hydronium ions in the substance being tested?
Nowadays you have to differentiate between pH measurements with non-glass sensors and with glass sensor. In the typical pH meter with a glass sensor, there could be many things that could go wrong resulting in biased results. In general, some electrodes may be sensitive to interfering ions such as $\ce{Ag+}$. So some potential could be generated also by other ions not only hydronium ions. However, the most common thing is that you are actually reading biased measurements because you are perturbing the status of your instrument.

Interaction of the solvent with the materials of your pHmeter

First thing you have to consider is the interaction of the solvent with the materials of your pH meter. You can easily damage some part of it or dehydrate the glass membrane of your electrode.

Solubility of the electrolyte in the junction

Then you have to bear in mind that in your glass pHmeter there is a junction that leaks an electrolyte for forming the saline bridge if this electrolyte is not soluble in your solvent the "circuit" will not close and you will have a wrong measurement.