The short answer is: No. There are no requirements on the system you want to model.
There are plenty of different flavors of Monte Carlo (MC) and the method you choose has to meet various conditions. The most common version of MC are Markov-chain Monte-Carlo methods such as Metropolis-Hastings. The two basic conditions are
Ergodicity: You have to be able to reach all relevant parts of the phase space by a) sampling long enough and b) choosing your steps such that there are no "forbidden" areas in the phase space.
The sampling weight for each state point has to converge to the Boltzmann distribution.
When detailed balance is fulfilled, you can prove that the Boltzmann distribution is a stationary (equilibrium) distribution. Technically, however, detailed balance is not a necessary condition.
To stress the first point: These are requirements on the Monte-Carlo method, not on the system. The most "general" Monte-Carlo method would be to sample random points in phase space and weigh them according to the Boltzmann distribution. This method tends to converge extremely slowly, but is not dependent on the conditions stated above.