If you are using ArcGIS, an option is the Erase tool
There are steps you can follow that I found clip hole in raster
Here are the steps... (Just swap out your layers accordingly.)
Create a vector polygon of the entire US (including Texas).
Use the "Erase" tool to clip out (erase) Texas from the US shapefile. Now you will have a polygon of the US with Texas removed.
Use the Extract By Mask tool and have the DEM as your input raster and the new vector shapefile of the US (with Texas erased) as the other input.
You will now have a DEM with everything but Texas.