Does salt hydrolysis takes place in buffer solution?
Water autohydrolyzes, or autodissociates, or autoionizes, to produce a very small amount of H+ and OH- ions. The product of their concentrations is 10e-14. 
Most salts that dissolve in water dissociate into ions. Often, one of these ions will have a strong attraction for H+ or OH- and tends to reduce the concentration of that ion in the solution. But since the autodissociation of water is an equilibrium process, if an ion (like acetate ion) combines with the H+, more water will dissociate to produce H+ and also OH-. When sodium acetate is added to water, it dissociates into Na+ and CH3CO2- ions; the Na+ ion attracts several water molecules around it without a preference for OH-. The CH3CO2- ion exhibits a strong preference for H+ over plain H2O, and so would reduce the H+ concentration in the solution. Water compensates by producing more H+ and OH- to keep their product equal to 10e-14.

Acetate ions continue to absorb H+ (it also has an equilibrium to maintain), and finally the acetate ion may be largely present as acetic acid (HCH3CO2). The extra OH- ions that came from the water hang around increasing the pH.

So, the answer to your question is yes, the salt hydrolyzes; in fact, the hydrolysis of the salt is what allows a buffer to be made. You establish a ratio of concentrations of the salt and its acid to give a desired pH. Instead of adding tiny amounts of acid to affect a tiny equilibrium product like 10e-14, you use the ratio of larger amounts of buffer ingredients, governed by the acid dissociation constant.

If the salt doesn't hydrolyze, like NaCl, which ionizes without changing the pH of water, it isn't useful as a buffer material - you can't use it to affect the concentration of H+ or OH- in water.