Why would a star’s position in the sky change relative to another star right next to it?

The answer is gravity.

Stars have a lot of mass so there for there gravity is strong.

If 1 star gets close to another star that has slightly more mass this would mean that it gets pulled in. As a result the orbit would change.

Edit

This does not affect all stars due to the distance they are from each other. Usually if it affects start it is in binary systems.

Thanks to Joan.bdm for this.