Will it hurt a Honda rancher 350 engine to run without a muffler?

I have a rancher 350 and I took the muffler off curious of what it sounded like and it sounded real good kinda like a Harley and I was wonder if it would hurt the valves or piston

On a highly tuned engine it's possible to rob yourself of a little power by shortening the exhaust and thus removing some engineered back-pressure. On an ATV you shouldn't have any issues, in fact you might get a little more power. The only consequence would be if your muffler had a spark arrestor in it. Removing it means they won't let you in a national/state park due to fire danger. Apart from that you're good to go!

How in the world does an ASE Certified Master Technician miss that removing the backpressure on the exhaust on this engine and not adjusting the carb mixture could seriously damage the engine? I'm calling BS on the first answer's credentials.

YES, it is possible to cause a problem running it that way. It isn't "guaranteed" to hurt it, but generally any adjustment to the intake or exhaust of a carbed engine without ensuring the resulting air/fuel mixture is still sufficient can cause the engine to run lean or rich. Rich isn't too big of a problem, more spark plug changes, carbon fouled components. Lean is a BIG problem, can lead to overheating and major engine damage. Guess which way removing the muffler (more air flow), and not adding more fuel pushes the mixture: lean.