That is a tricky one. I have often seen diagrams of how constellations change in time, especially The Plough (Big Dipper) and IIRC these suggest several thousand years at least. I guess the closer the stars involved the smaller distance you would have to travel, whereas very distant objects would probably appear pretty fixed.
The apparent location of stars within the field of view of an observer traveling through interstellar space would change with distance as the traveler got further and further from earth. The constellations, as we have defined them are, are made up of objects in vastly different positions in three dimensional space. Some are comparably close to us and others are extremely far away within a single constellation. The shape of a constellation would change depending on which one you were looking at.
Alpha Centauri is the closest star to us (it is actually a binary star) and is part of the constellation Centaurus. Traveling anything close to 4 light years tangentially to that constellation would alter the appearance of centaurus very quickly
There is a free program that will allow you to look at the stars from 'any' position in space and, therefore, see exactly how constellations change for yourself. See: Celestia: Home
At the velocities at which we would be traveling,any changes in the appearance of most of the constellations would be negligible.
Knowing the distances to the nearest stars, whether or not they were part of the constellations, would be much more useful in verifying the location of the traveler.