Detailed below is an example showing how a set of pierced faces can be fixed using the repair tools in the Surface Mesh Diagnostics dialog.
The Browse
buttons were initially used to select the area of interest, as shown below:

The offending pierced faces are colored red. When using the browse buttons, the area is automatically isolated and the faces being pierced, along with the edges of the piercing faces, are selected (highlighted) for you.
There are two approaches that could be used to fix an area such as this. One method is to use the Smooth Selected Vertices
button to try and move the offending vertex at the tip of the piercing back to the original surface. This will work in cases where the vertex at the tip is not connected (i.e. merged) to any other part of the surface.
A second approach is to simply delete the offending faces involved in the piercing and then re-fill the hole in the original surface. This is the approach we will use for this example.
To delete a face, all you need to do is use the cursor to click on the face. It will be selected as shown below:

With a face selected, the Delete Selected Faces
button will become active. Pressing this button will delete the selected face:

On deleting the face, new errors, namely free edges will be created. These are shown by the green colored edges that will appear on the model. These can be ignored for now. The remaining faces making up the piercing can now be selected (using the Ctrl key for multiple selections) and deleted using the same method as described above. The area will now look like this:

Note you could also have double-clicked on any of the piercing faces to select all of them, along with the edges of the pierced faces.
Again, we have green lines indicating free edges in the model. The good news is that there are no more red faces in the upper surface which used to be pierced. We are now dealing with simply a hole in one surface.
The easiest way of fixing a hole is to select the edges surrounding the hole so that a complete loop is formed. This can be done by pressing the Ctrl key and using the cursor to select the edges one at a time or alternatively, you can double-click on any edge next to the hole to select them all in one go:

The Fill Holes using Selected Edges
button will be active. This button can be pressed to automatically fill the hole with triangular faces:

The green lines will also disappear at this point indicating the problem area has been completely fixed. The quality of the triangles used to fill the hole can be improved by locally remeshing the surface. You can use the cursor to select the bad quality triangles in the surface (use the Ctrl key to make multiple selections).

Once the faces have been highlighted, press the Remesh Selected Faces
button to locally resurface the triangles.

The surface is now fixed and ready to be used for volume meshing.