The plane is beneath the observer. As long as light is travelling on straight lines, it cannot possibly appear above him, or above his horizon.
Yes, it extends away indefinitely, but things that ate further away also appear smaller. So every extra bit of groundplane he can see is progressively smaller than the last, so it only ever approaches the horizon, never stacks above it.
Basically, all that infinite real estate becomes more compressed into your finite field of view.
Yes, applying infinity to the real world leads to absurdities, but it doesn’t mean you’d still see the plane above your eye-level. I do imagine it would give you a strange sensation, as if the ground is seemingly “higher”, or that you feel as if you’re in a valley of sorts, because we’re used to the positive curvature of the earth falling away.
But in geometry it’s possible to have infinite space within a finite boarder. Or an infinite border within a finite space: Fractals.
The red plane the man is standing on stretches off to infinity. His eyes are 5ft above the ground. But any line of sight above the horizontal will never intersect that plane (lines shown in blue). Lines of sight below the horizontal (shown in red) will intersect the plane, at a distance x that is dependent on the angle a with the horizontal.
Simple trig, remember soh-cah-toa? Tan a = 5ft / x
or, x = 5ft / tan a.
So for an angle of 45 degrees below the horizontal, you are seeing (5ft / tan 45º) = 5ft away.
For 30 degrees, you are seeing 8ft 8in away.
For an angle of 10 degrees below the horizontal, 28ft 4in away
For an angle 1 degree off horizontal, about 286ft away
By the time you get to a thousandth of a degree off horizontal, you can see 54 miles.
To see 1,000 miles (5,280,000ft) you would have to look (arctan 5/5,280,000) = about 0.000054 degrees below horizontal.
And so on…
Actually, the simplest possible refutation of this is: the other infinite thing in the observer’s field of view is the empty space above the infinite groundplane - why would the infinite ground swallow up the infinite sky?
I think we’ve pretty much establish nothing will appear above the observers eye level. We also have the appearance of a horizon (although not a true horizon) at the observer’s eye level. For the plane to (appear to be) below the observer’s feet at point 0 and eye level at infinity, there should be a perceived curvature.
If there is no differentiation of distance (brightness, hue, sharpness etc.) then the plane would appear as an eye-high wall of an indeterminant distance away. If there were a way to distinguish distance as in the plane gets darker (brightness) or lighter (saturation), I would argue that the observer would think that they are in the bottom of a bowl shaped like half of a sinusoidal curve of infinite wavelength.
Is the plane on a treadmill?
Dont you start! We’ve just barely got the the OP under control here. 
Let’s just make this even simpler: Humans standing on a flat featureless plain on the Earth can’t see the curvature of the Earth. So as far as appearances to humans are concerned, what you’d see on an infinite plane would be just like what you see in Kansas.
What if there were snakes on the plane?
The answer is . . . ***anything. *** If the plane is truly infinite . . . if it literally never ends, in all directions . . . then it is infinitely larger than the Universe, and thus, infinitely older than the Big Bang. So it is outside the laws of identity and causality.
I still don’t think this is true.
Imagine standing in a large empty warehouse, 300ft wide, with your back to one wall. The floor is flat and is a contrasting colour to the wall opposite you, and the length of the warehouse (perpendicular to the direction you are facing) is considerably longer than 300ft, so you can’t see the corners without turning your head. As shown in the calculations in my earlier post, the level of the “horizon” (i.e. the point at which the wall meets the floor ahead of you) will be less than one degree below your eye level.
In this situation, I can’t believe that the floor will look “bowl-shaped”, so why should it suddenly look bowl-shaped if the horizon were less than one degree higher (as would be the case with the far wall an infinite distance away)?
If there is no apparent curvature, then how does the plane go from beneath your feet to apparent eye level?
Perspective. 
If you’re above a plane, then points further away will always look “higher” than points closer to you. Similarly, if you’re below a plane (e.g. a ceiling) then further points will appear lower. That doesn’t mean they look curved.
Try this - get a sheet of white paper and hold it flat, horizontally, with one end under your chin. The far end of it will look a lot closer to eye level than the near end, but it still looks flat.
Or, if you’ve got a large empty tabletop or desk to hand, rest your chin on it and look towards the far end. The same principle applies. The plane still looks flat.
Note that no matter how high you are above the ground, the “horizon” will still appear at exactly eye level.
So standing on the ground, the world may not look like a shallow bowl.
But now fly up to an altitude of 5 miles. Surely the world looks like a bowl now!
I think this is pretty much it. Imagine the following:
Let’s consider a function y=f(x) where x represents the distance (in Units) from you of any point on the infinite plane, and y represents the declination of that point (or the circle of points) that is that distance from you, calculating from your eyes. When x=0 (the point at the nadir directly under you), the point is 90 degrees down (y=-90). Use trigonometry (come on, you can do it!) to determine the angle (y) at various points as x grows to infinity (you know two sides of the triangle, which are x itself and the height from the ground to your eyes in Units. As you plot the function, the curve will asymptotically approach 0 as x grows without bound.
Of course you may substitute for Units whatever unit of distance floats your boat - meters, feet, parsecs, furlongs, light years, micrometers, whatever.
Feel free to measure angles with Radians if you feel like it.
Amended: Colophon beat me to it! See his “simple illustration” post.
At least we know that the plane is not in Eastern Europe.
There are no Poles!
As I mentioned above, we have an asymptotic approach toward the center of your vision when looking straight ahead.
It’s not a horizon in a geographical sense because it doesn’t curve down, but it does divide your vision.
Perhaps someone with some time could create a VR applet?
Wheat?
I don’t think we’re in Kansas anymore.
And if you look at one of my earlier posts, I said that one possibility was that the plane would appear flat like a wall.
