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We investigate linear and non-linear dynamics of spherically symmetric perturbations on a static configuration in scalar-tensor theories focusing on the chameleon screening mechanism. We particularly address two questions: how much the perturbations can source the fifth force when the static background is well screened, and whether the resultant fifth force can change the stability and structure of the background configuration. For linear perturbations, we derive a lower bound for the square of the Fourier mode frequency $ω^2$ using the adiabatic approximation. There may be unstable modes if this lower bound is negative, and we find that the condition of the instability can be changed by the fifth force although this effect is suppressed by the screening parameter. For non-linear perturbations, because we are mainly interested in short wavelength modes for which the fifth force may become stronger, we perform numerical simulations under the planar approximation. For a sufficiently large initial amplitude of the density perturbation, we find that the magnitude of the fifth force can be comparable to that of Newtonian gravity even when the model parameters are chosen so that the static background is well screened. It is also shown that if the screening is effective for the static background, the fluid dynamics is mostly governed by the pressure gradient and is not significantly affected by the fifth force.