Generating a C++ header containing OpenGL function definitions
In python it is not necessary to end a line with a ;.

Your function generate_header_file is probably better named write_to_header_file.

I would read the file only once and do everything in one pass in generate_header_string:

from itertools import chain

def generate_header_string(function_names_file_name):
    header_string1 = [INITIAL_HEADER_STRING_ONE]
    header_string2 = []

    def_string1 = "PFN{}PROC {};"
    def_string2 = "\t{1} = (PFN{0}PROC)GetGLProcPointer(\"{1}\");"

    with open(function_names_file_name, "r") as function_names:
        for line in function_names.readlines():
            line = line.rstrip("\n")
            line_upper = line.upper()
            header_string1.append(def_string1.format(line_upper, line))
            header_string2.append(def_string2.format(line_upper, line))
        header_string1.append(INITIAL_HEADER_STRING_TWO)
        header_string2.append("}")
        header_string2.append("\n#endif")

    return "\n".join(chain(header_string1, header_string2))


Here I changed the accumulation variable to lists which I join in the end. This gives some memory boost, because string addition is very costly, especially if the strings become bigger (and your INITIAL_HEADER_STRING_X is already quite big). The lists also get rid of having to manually write \n in the strings, because we can just use it as joining variable at the end.

You will want to give the two variables better names (like header_declaration and header_initialization or whatever fits their use case; I don't know any OpenGL syntax, so I'm not exactly sure).

I also declared the strings you format once before the loop to avoid having to redeclare them every iteration. They could also do with better names, probably something similar to the header lists.