I'm following an OpenGL tutorial that uses a certain input/window handler (i.e. GLUT, GLFW), but I cannot, due to platform issues, use that handler. Can I use the exact code from the tutorial despite my using a different input handler? Does the OpenGL code have to be modified to work with a different handler?
It will work.
OpenGL is completely agnostic of input. It is a graphics library, and as such, cares only about graphics. Everything else, including input, audio, and all else, is completely and utterly irrelevant.
The only difference for you is that toolkits like GLUT, GLFW, SFML, and others do the setup of an OpenGL context for you.
If you want to use another toolkit, that's fine, and it will probably also set up your context for you. You can also use OpenGL directly, in which case, you will need to create the context yourself, which will require calling into the WGL (Windows), AGL (Mac), GLX (X-Windows on *nix), or EGL (everything else) APIs.