The natural frequency of an object or volume of a material depends only on the dimensions of the object and the material it is made of.
Therefore, if the natural frequency of an object were to vary with waves having different frequency, amplitude, or wavelength, It must mean that the property of the material should change on passing a different wave through it.
As Carl Witthoft said :
If the wave passes through without attenuation* (or redirection) then there's no energy transfer and almost impossible for the material to change property w/o changing its entropy. If some of the wave's energy is absorbed, then lots of things can happen.
And as most materials, liquids or gases have either viscosity or a modulus of rigidity, zero attenuation of sound is impossible due to the second law of thermodynamics**.
Thus, it's acceptable to say that all objects change their natural frequency by some or the other amount when a wave passes through it.
Also, the change in the natural frequency might be different for waves having different amplitude, wavelength or frequency.
To know exactly which materials vary in what way with change in property of the wave, visit engineering.se .
*Definition: Acoustic attenuation is a measure of the energy loss of sound propagation in media.
**As per the second law of thermodynamics, during any process (i.e travelling of the sound wave) occurring in a closed system (i.e the object through which the wave passes and that whose natural frequency is being measured), the entropy of the system always increases (and therefore attenuation always occurs).