This is easier than it seems:
Raster -> Conversions -> Polygonize
Vector -> Geoprocessing Tools -> Clip
Depending on your raster file format, the options to polygonize are different, but commonly you choose "DN" as attribute. And if your raster has a proper mask layer (or NaN values), you can select that so these are omitted, too!
Et voilà, you get a simple vector file that traces the outline of your raster.
Then, use the Clip tool: Choose the fault polyline layer as input, and the polygonized raster layer as Clip layer.
Keep in mind that the polygonized raster will never be a perfect fit to the raster file. So depending on your use case, you might want to manipulate it (buffer +-) to ensure no polylines are cut off too short.