It shouldn't, if the devices are engineered properly to not be affected by small fluctuations in line frequency.
In the simplest case, a lightbulb (or any other resistive heating element) has a power consumption which is independent of the line frequency.
However, this scenario couldn't really occur in the case of a single isolated power company, since AFAIK the national grid (at least in the US) is set up so that generators are all phase-locked to 60 Hz. Otherwise, if there was a phase drift or (worse) any frequency difference with one particular supplier, there would be catastrophic problems when the company attempted to supply energy to the grid.
In practice, it is actually the frequency (or more specifically, the phase) which deviates slightly in response to load, rather than load deviating in response to deviations in frequency, as mentioned in this question.