I just shot a bunch of eyeball inserts for my last movie using a 200mm macro lens about a foot away from the actor.
I've done it before with the long end of the 24-290mm Optimo using a diopter, usually again with maybe the front of the lens only about a foot and a half from the actor.
So first of all consider the proximity of the camera body and lens itself to the actor and the fact that the camera will take up the center of the reflection in the eyeball in a front angle, and consider that the lighting unit will also be reflected. Lately I've been using a small hard source just because it's less distracting to have a bright dot reflected than a large white panel from a soft light, unless very much from a side angle. But it also depends on what I consider the "real" reflection of the source to be, a window, a lamp, a fire, etc.
So you could shoot an insert of an eyeball, not super tight, at the 100mm end of your lenses with a diopter but you are going to be a few inches from the face with the lens.
As far as reflecting fire, I have a shot like that in the movie I just shot and we are doing it as a post composite.
And I also had a shot like that in "90 Minutes in Heaven" -- I screen-grabbed this off the trailer that's online so it's rather compressed, but here is the brief flash cut to the eyeball:
During a crash sequence, I shot a truck driving right into a GoPro (which has a natural wide-angle distortion) head-on. I shot the eye with the 290mm end of the Optimo zoom with a diopter. You can still see the bright white dot of the light used to light the eyeball, which I guess could have been painted out but it's a flash-cut.
You can also see the chromatic aberration from the diopter, which is why I used a 200mm macro on my next movie to shoot the eyeball shots. And you can see the focus problems I had even at around f/8-11 at 290mm.
The problem is that a real eyeball is like shooting a chrome/silver Christmas Tree ball ornament. i.e. the reflection is like a near fish-eye lens shot of what the eye is facing, being a curved surface. It sees the world and it also shrinks everything down into a small reflection. So the problem with reflecting a TV set or projection screen is that it would have to be gigantic and close to the eye to not see the squared edges of the screen. Second problem is that the screen is a flat image and a real reflection would be distorted and curved more.
So I think you have two options: shoot a real fire reflected in the eyes but shoot from a more side-angle so that the person can face the fire and the camera can get out of the way of blocking the reflection, or do a composite so that the reflected element can first be distorted into a fish-eye type circular shape before being overlaid over the eyeball.
Here are two examples:
The first, from "Blade Runner" is interesting because the eyeball itself is a still photograph, it doesn't move, making it a lot easier to do a lock-off composite. You'll find that with macro shots of people's eyes, even the slightest head movement is really bad for framing and focus, you practically have to clamp the head down or at least, get them to firmly press their head against a wall or something stable. And you need a lot of light to stop way down, and even then, you'll have to decide if you want the surface of the eye to be in focus, or the iris of the eye, which is deeper, you can't hold both even at f/16.
So for "Blade Runner", the reflection was one of their landscape shots with the fireballs going but projected onto a round white dome to create a fish-eye distortion before being composited over the eyeball frame. Today there are plug-ins that will create a fishbowl distortion to a shot with a lot of vignetting.
The second shot is from "Apocalypse Now" and is a real fire being reflected on a real eyeball but from a profile angle.
If you do a composite in post, then there is no real reason why the elements have to be shot with an anamorphic lens.