I don’t think this is possible with Dynamo. The reason is that you need an active/open document to run Dynamo and these errors happen when the document is opening, but has not opened (there’s a difference).
If you are really keen to implement something like this, I suggest you have a look at or learn the Revit API, specifically document events like the OnDocumentOpening event and the Failure API where you can create your own failure handler. Jeremy Tammik covers both of these in detail on his blog…
However, some of these errors are important, and can’t or should not be dismissed, fixing the cause it always a better solution if you can help it - for instance having good practices like not linking cad files from a local machine to a workshared model.