If you are creating large arrays, then it can save a considerable amount of memory to use narrower types (less bytes), as the size of the array will be "type width" * "number of elements" + "overhead".
However, I'm pretty sure by default that in classes and structs, they will be packed along whole word boundaries e.g. 32bit = 4bytes. A short will still be packed into a 4 byte space.
You can however, manually configure packing in structs\classes by using structure layout:
http://msdn.microsoft.com/en-us/library/system.runtime.interopservices.structlayoutattribute(VS.71).aspx
As with any performance related issue: "don't think - measure".
From an API perspective, it can be majorly annoying to have to keep casting from shorts to ints, etc, as you will find most APIs will use ints for example.