Do I need to compile the header files in a C program?
Firstly, in general:
If these .h files are indeed typical C-style header files (as opposed to being something completely different that just happens to be named with .h extension), then no, there's no reason to "compile" these header files independently. Header files are intended to be included into implementation files, not fed to the compiler as independent translation units.
Since a typical header file usually contains only declarations that can be safely repeated in each translation unit, it is perfectly expected that "compiling" a header file will have no harmful consequences. But at the same time it will not achieve anything useful.
Basically, compiling hello.h as a standalone translation unit equivalent to creating a degenerate dummy.c file consisting only of #include "hello.h" directive, and feeding that dummy.c file to the compiler. It will compile, but it will serve no meaningful purpose.
Secondly, specifically for GCC:
Many compilers will treat files differently depending on the file name extension. GCC has special treatment for files with .h extension when they are supplied to the compiler as command-line arguments. Instead of treating it as a regular translation unit, GCC creates a precompiled header file for that .h file.
You can read about it here: http://gcc.gnu.org/onlinedocs/gcc/Precompiled-Headers.html
So, this is the reason you might see .h files being fed directly to GCC.
Okay, let's understand the difference between active and passive code.
The active code is the implementation of functions, procedures, methods, i.e. the pieces of code that should be compiled to executable machine code. We store it in .c files and sure we need to compile it.
The passive code is not being execute itself, but it needed to explain the different modules how to communicate with each other. Usually, .h files contains only prototypes (function headers), structures.
An exception are macros, that formally can contain an active pieces, but you should understand that they are using at the very early stage of building (preprocessing) with simple substitution. At the compile time macros already are substituted to your .c file.
Another exception are C++ templates, that should be implemented in .h files. But here is the story similar to macros: they are substituted on the early stage (instantiation) and formally, each other instantiation is another type.
In conclusion, I think, if the modules formed properly, we should never compile the header files.