A production is a living thing that changes on a day-to-day basis. Though a script is the base skeleton of any film project, the director needs to be responsive to changes that may occur on set or alterations with his or her vision. It is important to limit these and know when they should happen to make them successful.
Many changes in the script only occur because interferences are so strong that they alter the shooting schedule permanently. For example, if a location pulls out entirely and you cannot retain anything similar then the script is going to have to be reworked somewhat so that you can include an available location. This usually is not a large thing unless said location, or other element, is crucial to the story. The is also unavoidable in situations where the script must be changed, so it is good to create back up plans ahead of time for places where your screenplay is dependent on things that have the ability to be pulled out. This means creating options for prop and location changes in the script during production that would maintain the themes and plot.
Since you as the producer and/or the director are not the only person working on this film other people may have input. Actors may want to suggest different lines or alterations to their scenes because they have a unique point of view on their character. Art people may have a suggestion about what would be more appealing in a location or correlative event. Be open to these suggestions, but the final say is still always up to you. In these situations its important to remember that film is a collaborative medium that will require you to make compromises based on the idea that creative professionals in other departments are going to be the best voice.
Once you see how things are playing out you may want to alter things yourself. You may see how an actor works with a scene and then want it to go a different direction. You may like a location so much that you want to spend more time filming there. There are so many things going on that it is inevitable that you are going to want to add things spontaneously. These will not always take the form of an actual script change if it is happening on the set, though if substantial changes occur during rehearsal you may want to include it. This is going to be especially true if it includes a change in lines or any type of substantial scene changes. The rest of the crew, especially the camera and sound department, are going to need to premeditate the actions and words that occur. The way this is communicated to them is through the screenplay, and changes need to be announced. It is, however, usually good enough to just give a little bit of notice and include them in the on site sides. If the changes is substantial enough then you are going to need to redo the storyboards.
All script changes need to be made known to the unit production manager and the script supervisor. These people need to record all changes, including spontaneous line changes by actors, so that they can be written for post-production editing and possible dialogue looping. It is natural to have these changes and they need to be accounted for so that everyone in the production knows how to alter their part accordingly. Organization is always the key in film and video production, and that is true even during spontaneous moments.