Should the gaps around the door casing be filled?
If it is an interior door - the answer is no.   Leave it as.



If it is an exterior door then yes for sure you should you should fill the cracks with insulation.   For anything big enough - over 1/4" maybe - I stuff Roxul insulation in.   Cram it in the gaps really good.    

Then once you have the bigger gaps at least as filled as you can get with standard insulation you spray the smaller gaps and the outside parts of the larger gaps with spray foam insulation (cheap big box example is Great Stuff - not a recommendation just an example, however I do recommend Roxul above).   Wait at least an hour - depends on temperatures - for spray foam to expand and slightly harden.   Then scrape off excess foam sticking out.    It is always good practice to get up close on each side and make sure you cannot see to the other side.  

An added step that I have seen in cooler climates is to (flashing) tape the "cracks" after insulation.   This for sure adds a wind barrier to the gap and might help with drafty doors/windows.