We theoretically evaluate the impact of drift-free noise on the dynamics of PT -symmetric non-Hermitian systems with an exceptional point, which have recently been proposed for sensors. Such systems are currently considered as promising templates for sensing applications, because of their intrinsically extremely sensitive response to external perturbations. However, this applies equally to the impact of fabrication imperfections and fluctuations in the system parameters. Here we focus on the influence of such fluctuations caused by inevitable (thermal) noise and show that the exceptional-point eigenstate is not stable in its presence. To this end, we derive an effective differential equation for the mean time evolution operator averaged over all realizations of the noise field, and via numerical analysis we find that the presence of noise leads to exponential divergence of any initial state after some characteristic period of time. We therefore show that it is rather demanding to design sensor systems based on continuous operation at an exceptional point.