Both the dopesheet and the graph editor show a red line when the data path for the fcurve cannot be found. While most times the fcurve stays connected to the correct item, there can be times when this doesn't happen, most often when the item is renamed or deleted in an external file from the one containing the animation. You will also notice that the shown data path is the complete raw path when it is broken, rather than the more user friendly abbreviation.
If an item has been renamed, we can correct the data path in the fcurves properties panel in the graph editor. If the item was deleted then you can just delete each fcurve, you can do that by selecting it in the list and pressing X or ⌦ Delete.
For a large number of items, it may be easier to make a script that will correct the data paths.