What effect would Mach 3 have on the human body?

I'm assuming you mean without an airplane! There'd be two major problems moving at that speed:

1. Air resistance - as air flowed over the human body at Mach 3, it would probably leave you rather burned or at least raw and bruised.

2. Breathing - you'd probably rupture your lungs trying to breath in air moving that fast.

It's possible that all you'd need is a protective suit and breathing gear.

Speed (as long as the ride is smooth) has no effect on the body, as all inertial frames are relative and therefore at rest in their own frame. I think you are confusing speed with acceleration. High accelerations (often called g-forces) can crush the body and drain blood away from the brain. The Apollo 10 astronauts traveled at Mach 33 relative to the earth with no ill effects. Now if you are considering traveling at these high speeds in the atmosphere, than obviously the aircraft has to be able to withstand the pressure and temperature from the air resistance. But the humans inside a properly designed airplane won't feel any effect of traveling at such high speeds.

An additional effect of supersonic travel is control reversal - where the resultant acting lift can move back, behind the aerodynamic centre of the wing. Therefore control inputs will produce the opposite response to that intented - a few test pilots were killed as a result of this phenomenon until understanding of the physics behind it was attained. Nowadays, modern Flight Control Systems will automatically "reverse" the controls if appropriate at these speeds, thus still allowing the pilot to fly care-free. Oh, and dont forget that if you were to overfly land at Mach 2 - the resulting pressure waves from the Mach cone would shatter most of the glass in the area.........

Mach 3, or any other Mac number, will have zero effect.

When you are moving at a steady speed you can't tell. For example when you are in a jet plane, providing it's not bumpy, you can't even tell you are moving unless you look outside.

It is extreme acceleration that can causes damage.

Humans can travel at any speed they wish to, provided the acceleration to that speed is within the limits of gravitational forces that put stresses on the body.