I currently use gdalwarp to clip a GeoTIFF file using a single shapefile. However, I would like to have the option of creating a clipped GeoTIFF file using multiple countries together.
Is using GDAL the proper way to approach this?
If the countries are adjacent to each other you could first dissolve (see https://gis.stackexchange.com/a/85046/15183 ) using ogr2ogr and then use the dissolved polygon for clipping.
You can merge all cutline geometries in single shape file in one or separate features. Additionally you can use such command line options:
For details see http://www.gdal.org/gdalwarp.html