I have made a script somewhat similar to this one: http://answers.unity3d.com/questions/5020/searching-a-project-for-missing-mono-script

But rather than just finding the null entries, I simply want to sweep through and delete them all. By that I mean remove the null references from the GameObjects's list of Components.

The documentation for GameObject.AddComponent notes that there is no such thing as RemoveComponent, requiring you to call Object.Destroy() to remove that entry. This is impossible because there is no object to Destroy.

Creating a new GameObject and adding the components of the old one to it would invalidate any references to it.

Is there a way to do this without laboriously clicking through the UI?

NEW EDIT: (in light of comments below)

Are these references to some funny objects representing a missing script that the overloaded == operator is treating as equal to null? No, because calling DestroyImmediate() does nothing, and trying to actually de-reference the pointer causes a null reference exception. It must be that there are null pointer entries in the list, and the list is not modifiable by editor script.

Here is the demonstration code:

[MenuItem("Custom/Remove Missing Script Entries")]
public static void Do() {
    foreach (Transform t in Selection.transforms) {
        Debug.Log(t.GetComponents(typeof(Component)).Length);
        foreach (Component c in t.GetComponents(typeof(Component))) {
            if (c == null){
                Debug.Log("NULL");
                // throw caution to the wind and destroy anyway!!! AHAHHAHAHAH!!!
                GameObject.DestroyImmediate(c);
                // awwww nothing happened.  still there.
            }
            else
                Debug.Log(c.GetType());
        }
    }
}