Do i need recalculate uv coordinate after modifying mesh vertices

Without more details on the exact effect you want to achieve, all we can say is "maybe".

If you want the texture to be sloshed around and stretched & squeezed sideways by the motion of the verts (say, for image distortion effects), leave the UVs as they are.

If you want the texture to keep its local structure/position the same as the shape appears to move under it (like a cat moving around under a printed sheet), you would want to modify the UVs in a way that counters the vertex motion in its tangent plane.

Still other effects will have other needs. Got a mock screenshot of what you want to achieve?

Great examples for why we need more information.

I can add that the easiest way to alter the UVs like in D$$anonymous$$G's second example is probably to use projective texture mapping, i.e. use a projector.