The intestinal epithelium in vertebrates has a characteristic architecture of protruding villi and receding crypts that enables nutrient absorption and cellular turnover. Intestinal organoids recapitulate its development and can be used as disease models, but the underlying mechanical processes are not fully understood yet. Here we combine advanced image processing and the bubbly vertex model for epithelial cell shape to achieve a fully three-dimensional reconstruction of cell shapes and forces during the development of mouse intestinal organoids. We show that the transition to budded morphologies is caused by a global increase in apical tension, which however is not maintained after budding, suggesting ratchet-like non-reversibility. We further demonstrate that luminal pressure decreases and basal line tensions increase during development, thus facilitating budding on the apical side, but at the same time mechanically stabilizing the system at the basal side, for example against cell extrusion. Our approach demonstrates how one can achieve a complete mechanical analysis of a complex tissue-like system.