I hate to provide an answer to my own question, but I wanted to share what I've found.

It seems my error here is that I assumed GUI.tooltip would be persisted outside of the OnGUI function. This does not appear to be the case, so I don't think we can rely on it in an Update() function, much less in another class.

My current solution is to use the following classes: A static GUIManager class to save the state of whether the mouse is currently over a GUI element. In any GUI Script, add some code to check the Tooltip (or hit test a texture or whatever code you want), and set the global static variable on my GUIManager. Using this approach, I can then check to see whether GUIManager.MouseOverGUI == true when determining whether I want to raycast or not. If anyone sees a flaw, feel free to contribute.

EDIT: Fixed the code to work with scenes that have multiple scripts implementing OnGUI(). Be sure to attach GUIManager to your GUI object so LateUpdate() gets executed to reset the MouseOverGUI bit prior to the OnGUI() logic.

public class GUIManager : MonoBehaviour
{
    public static bool MouseOverGUI = false;

    void LateUpdate()
    {
        MouseOverGUI = false;
    }
}

public class TestScript1 : MonoBehaviour 
{   
    void OnGUI()
    {
        Rect boxRect = new Rect (10, 350,150,200);
        GUI.Button (boxRect, new GUIContent("Box Header", "BoxTooltip"));   

        if (GUI.tooltip != "")
            GUIManager.MouseOverGUI = true;
    }

    void Update() 
    {
        if (Input.GetMouseButtonDown(0) && !GUIManager.MouseOverGUI)
        {
            print("got here.");
        }
    }
}