The resilience of biodiverse forests to climate change depends on an interplay of adaptive processes operating at multiple temporal and organizational scales. These include short-term acclimation of physiological processes like photosynthesis and respiration, mid-term changes in forest structure due to competition, and long-term changes in community composition arising from competitive exclusion and genetic trait evolution. To investigate the roles of diversity and adaptation for forest resilience, we present Plant-FATE, a parsimonious eco-evolutionary vegetation model. Tested with data from a hyperdiverse Amazonian terra-firme forest, our model accurately predicts multiple emergent ecosystem properties characterizing forest structure and function. Under elevated CO2 conditions, we predict an increase in productivity, leaf area, and aboveground biomass, with the magnitude of this increase declining in nutrient-deprived soils if trees allocate more carbon to the rhizosphere to overcome nutrient limitation. Furthermore, increased aboveground productivity leads to greater competition for light and drives a shift in community composition towards fast-growing but short-lived species characterized by lower wood densities. Such a transition reduces the carbon residence time of woody biomass, dampening carbon-sink strength and potentially rendering the Amazon Forest more vulnerable to future climatic extreme events.