Here's an old post I made on the subject,
Speaking from my experience with Mocha, it most definitively can. Perhaps the most tangible example of how rendering is a separate concern from game logic, even with unmodified vanilla Doom.exe, is to reason on how the automap works: "3D" rendering is effectively turned off (and all the slowdown that goes exclusively with it) while only the automap code "draws" on the screen, in what to some people appears like an asteroids clone ;-)
I'm curious: does compiled Linuxdoom run successfully if you manage to strip it of all the graphics, sounds and player input overhead, remaining a mere console utility instead? Keep in mind that the game logic must still work fully, but nothing to display other than relevant text messages at the DOS prompt (no video mode change, no waiting, no music, no sound).
Doom is quite neatly separated into concerns on its own right, and if you wanted to make e.g. a turn-based text-based adventure or roguelike out of it (by typing commands like "go forward", "turn right 30", "switch weapon 3" etc. it would be perfectly feasible. Heh imagine if instead of seeing the monsters, you only got a textual description of what objects are in your field of view and/or in the same room as you ;-)
Now, there are a few murky points in the code (e.g. some R_ functions that are called elsewhere, too) but upon analysis most of them turn to be pure, stateless functions, so they could just as well be placed in a "package" of their own.
Similar considerations can be done for sound and player input: sound can end up into "dummies" without changing the code itself (there's s_sound.c, which is gameplay-specific and only does high-level positioning and channel management, and i_sound.c which is system-specific and actually does the gory mixing/DSP writes. You can have one without the other, if so you wish. You can make all functions of s_sound.c be dummies, or having s_sound.c completely functional but have no system-specific sound code (that's the approach I used for Mocha while it still had no sound).