Adding inequality constraints (e.g. positivity, monotonicity, convexity) in Gaussian processes (GPs) leads to more realistic stochastic emulators. Due to the truncated Gaussianity of the posterior, its distribution has to be approximated. In this work, we consider Monte Carlo (MC) and Markov Chain MC (MCMC) methods. However, strictly interpolating the observations may entail expensive computations due to highly restrictive sample spaces. Furthermore, having emulators when data are actually noisy is also of interest for real-world applications. Hence, we introduce a noise term for the relaxation of the interpolation conditions, and we develop the corresponding approximation of GP emulators under linear inequality constraints. We demonstrate on various synthetic examples that the performance of MC and MCMC samplers improves when considering noisy observations. Finally, on 2D and 5D coastal flooding applications, we show that more flexible and realistic emulators are obtained by considering noise effects and by enforcing the inequality constraints.