As the question states, do vibrations cause any influences on inductive sensors? For example, do vibrations on rail, from a locomotive, affect signalling equipment?
I can't tell about railway sensors, but for automotive (traffic lights).
Where it comes to reliability of readout - "false positives" - no, they don't.
The sensors are inducing a pretty high own frequency (which means any naturally occurring mechanical vibrations are going to be at worst a distant harmonic), and perform a lot of post-processing after detection of frequency as caused by changed inductance.
Both timing and intensity of the change are taken into account. As for timing, both too short pulses (EMI noise, surges, semi-distant lightnings etc) and too slow changes (weather, temperature due to time of day) are ignored; the long ones are integrated into the baseline to compare against too. Similarly, too weak shifts are ignored even if their timing is right - to avoid false positives of vehicles in neighboring lanes.
How do the vibrations affect the inductive sensors? Happening for long enough, with sufficient intensity, they cause material wear and break the induction loops, necessitating repairs. This is the primary disadvantage of inductive sensors and by far the most frequent mode of failure. Luckily such damage is easily detected by the circuitry, so the sensors can alert the services, fall back to backups, or enter "failsafe mode" e.g. producing "always active" signal, resulting in non-optimal but acceptable control, or switching the controller to a failsafe "fixed time" program.
Coils by themselves are not susceptible to mechanical vibrations. However, relative motion between a coil and a external magnetic field can cause a signal out of the coil. This is because the changing magnetic field at the location of the coil causes a signal. After all, what coils sense is the change in the magnetic field around them.
If the coil and the source of the magnetic field were to vibrate together, then there would be no signal from the coil. That is because relative to the coil, the magnetic field is not changing.