Let’s see… Once you reach about 30 mph, your leg muscles reach a limit of how fast they can contract and release, and how much power they can produce to overcome the air resistance of your body.
If you ignore that piece of biology, when the wind in your face exceeds 50–60 mph, your eyes start watering. That’s why most motorcyclists wear eye protection even if they don’t believe in helmets.
If you ignore that piece of physics, this is what happens to your face if you’re running at just 457 miles per hour. Somehow, I think it would be tough to see anything when you have your eyes squeezed shut as tight as possible to keep your eyelids from flapping back and forth in the wind:
If you ignore that piece of physics, consider what happened to one of the few people alive who survived a blast of supersonic air when he ejected from a fighter jet:
Air Force pilot Capt. Brian Udell is one of the only pilots in history to survive after ejecting from a fighter at supersonic speeds. The force of the air moving at more than 768 mph on his body was so strong that it nearly killed him.
“It felt like somebody had just hit me with a train,” said Udell. “When I went out into the wind stream, it ripped my helmet right off my head, broke all the blood vessels in my head and face, my head was swollen the size of a basketball and my lips were the size of cucumbers. My left elbow was dislocated and pointed backward, the only thing holding my leg on was an artery, the vein, the nerve and the skin and my left leg snapped at the bottom half.”
His body was essentially being torn apart by the wind.
This US Air Force pilot ejected while flying supersonic (and survived)
If you ignore that piece of physics, somewhere around 10,000mph you burst into flames from the heat generated by the air compression in front of you. See any re-entry from space to see how that works out.
If you ignore that piece of physics, you have to deal with the fact that somewhere around 17,000mph you’ve reached orbital velocity and are no longer running, as you’ve left the surface of the earth. Hope you know how to run that fast in a space suit.
If you ignore that piece of physics, you have to deal with the fact that once you get up to actual high speeds, you need to have acquired kinetic energy. A lot of it. Turns out that just 1 kilogram of mass, moving at just 0.5c, has about 10 megatons of energy. Multiply by your weight in kilograms to find out how much more energy it takes to get your entire body up to just half light speed… Figuring out how your leg muscles release that much energy without you turning into a very large mushroom cloud is left as an exercise for the student. The Tsar Bomba released about 50 megatons. You probably weigh more than 5 kilograms.
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If you ignore that physics and start running even faster, at speeds where relativity actually makes a difference, you discover that trying to stay on the surface of the earth generates a lot of centrifugal force. Hint - if you’re moving at just half lightspeed around the sun at the distance of Earth’s orbit, that’s around 100,000Gs. Shrink the turning radius from 93 million miles down to 8,000 miles, and the centrifugal force goes up another factor of 12,000 or so…
If you ignore that physics and keep running faster, you discover the wonders of Cherenkov radiation. Although nothing can exceed the speed of light in a vacuum, light slows down in materials - to about 0.992c in air, or 0.66c in water (the ratio of the speed in vacuum and speed in the material is called the index of refraction). Now it turns out that a particle can exceed the speed of light in a given material - but you start radiating away energy until you drop below the speed of light in the medium. That’s called Cherenkov radiation - it’s the cause of that weird blue glow around spent nuclear reactor fuel in water-filled storage tanks. But once you’re over 0.992c, you’re going to be radiating off a lot of kinetic energy (which you’ll need to keep replacing just to maintain speed). And you probably won’t be able to see anything through the bright glow of the Cherenkov radiation in front of you - in fact, it’s almost certainly going to be bright enough to fry your retina and optic nerve.
If you ignore that physics and keep running faster, you discover that the energy requirements to accelerate further keep going up - to the point where you take all the matter in the visible universe and convert it to energy, and it’s still not enough to actually get you up to the speed of light in a vacuum, much less past it.
If you ignore that physics, you discover that all the oncoming photons have been drastically blueshifted, to the point where even the longest wavelength radio waves now appear in the ultraviolet where your eyes won’t register it. If you were able to look back, even gamma rays would be redshifted to invisibility. But you can’t look back, at these speeds you need to watch where you’re going…
If you ignore that physics and actually reach the speed of light in a vacuum, you’re now a cloud of photons, because no particles with mass can travel at the speed of light in a vacuum.
If you ignore that physics and keep running faster and actually exceed the speed of light, you have another problem - time just went sideways. Literally. A lot of people think time goes backwards if you exceed the speed of light. It doesn’t - it goes at right angles. You probably don’t have eyes anymore, because what used to be the protons, neutrons, and electrons of you have all changed into weird unknown particles that behave differently and have shot off in all directions.
If you ignore that physics… well… yes, you can see. However, since you’ve thrown out basically all of the physics in our universe, you’re obviously in a universe with completely different laws of physics… and one of them is that exceeding the speed of light is punished - by turning the violator into either a My Little Pony or a Smurf for the rest of their natural lives.