Shape-memory polymers are of high scientific and technological interest in the biomedical field, e.g., as matrix for self-anchoring implantable devices. In this study, two different star-shaped copolyester tetroles, semi-crystalline oligo[(-caprolactone)-co-glycolide]tetrol (oCG) and amorphous oligo[(rac-lactide)-co-glycolide]tetrol (oLG), were synthesized and subsequently crosslinked by a low molecular weight diisocyanate resulting in copolyester urethane networks (N-CG, N-LG). Both networks could be loaded with model drugs and a diffusion controlled release of the drugs was observed without any effect on the mass loss as measure of hydrolytic degradation. However, the N-CG network’s capability of shape programming was disturbed as the crystallinity of the precursors got lost in the complex three dimensional architecture after crosslinking. By contrast, amorphous N-LG network showed an excellent shape-memory capability with a switching temperature around 36 °C corresponding to their glass transition temperature. This led to triple-functional materials combining biodegradability, shape-memory, and controlled drug release.