Origami, the art of paper folding, has long fascinated researchers and designers in its potential to replicate and tap the complexity of nature. In this paper, we pursue the crossing of origami engineering structures and biology, the realm of biologically-inspired origami structures categorized by the two biggest taxonomy kingdoms and DNA origami. Given the diversity of life forms that Earth comprises, we pursue an analysis of biomimetic designs that resemble intricate patterns and functionalities occurring in nature. Our research begins by setting out a taxonomic framework for the classification of origami structures based on biologically important kingdoms. From each of these, we explore the engineering structures inspired by morphological features, behaviors, and ecological adaptations of organisms. We also discuss implications in realms such as sustainability, biomaterials development, and bioinspired robotics. Thus, by parlaying the principles found in nature's design playbook through the art of folding, biologically inspired origami becomes fertile ground for interdisciplinary collaboration and creativity. Through this approach, we aim to inspire readers, researchers, and designers to embark on a journey of discovery in which the boundaries between art, science, and nature are blurred, providing a foundation for innovation to thrive.