A change in pH caused by the reaction could affect the conductivity by affecting the ionization state of any solutes. Also, the products of the urease reaction (ammonia and carbon dioxide) are both volatile, so they could evaporate from the solution. Stirring could influence the rate of their evaporation. I would keep the stirring rate to a minimum.

I do not think the magnetic stirrer would affect the conductivity readings, as they are almost always used together. Do you observe a change in the conductivity reading if you swirled the solution by hand (away from the stirrer) versus using the magnetic stirrer? This would be best done with a mock solution, with no urease and a known amount of NH4+, so the concentrations are not changing.  Similar methods have been used to quite accurately measure urease reaction rates (and screen for inhibitors) by monitoring pH, or using an ion specific (NH4+) probe. The very useful BRENDA site should have some inhibitors to try, once you work out the measurements.

Magnetic stirring won't interfere your measurements. Urea hydrolysis reaction is very fast ( in terms of minutes only with respect to pH also). 

A change in pH caused by the reaction could affect the conductivity by affecting the ionization state of any solutes. Also, the products of the urease reaction (ammonia and carbon dioxide) are both volatile, so they could evaporate from the solution. Stirring could influence the rate of their evaporation. I would keep the stirring rate to a minimum.

you can turn off the stirrer and test its effect or test its efect  by measuring conductivity of stable solution with and without stirrer.