Can helicopters fly a loop?

Yes, some helicopters can indeed do loops, barrel rolls and even hammerheads … although it takes a specifically designed ‘rigid’ main rotor system. Such rotor systems are dominantly found on military-type helicopters with the exception of a few civilian helicopters.
Robinson R22  

The R22 is a classic example of design excellence. No other helicopter in its class can match its speed, reliability, affordable price, and low operating costs. The R22 has fewer accidents due to aircraft or engine failure than other light helicopters, and Robinson's commitment to the highest quality in design, materials, and craftsmanship make it the enduring choice for flight training, radio traffic watch, power line patrol, and other applications that require performance, dependability, efficiency, and easy maintenance.

R22 Beta II