As one of the most popular 3D printed metamaterials, the auxetic structure with its tunable Poisson’s ratio has attracted huge amount of attention recently. In this study, we designed an auxetic shape-memory metamaterial, which showed designable buckling responses by using the thermomechanically coupled in-plane instability. The influence of viscoelasticity on in-plane moduli and Poisson’s ratios of shape-memory auxetic metamaterial was experimentally investigated. Based on the generalized Maxwell model and finite-element method, the buckling behaviors and their main influence factors were studied. The analytical results and experimental ones showed a good agreement. Thermomechanical properties of the printed metamaterials govern the temperature and strain rate-dependent buckling, and a controllable transition from the negative to positive Poisson’s ratio in the metamaterials can be achieved. Based on the shape memory effect, the buckled state and the Poisson’s ratio of the metamaterials can be tuned by programmed thermomechanical processes. This study provides a simple and efficient way to generate morphing structures using the designable buckling effect.