This is false for several reasons. Shrubs and trees store carbohydrates (food materials) in their branches and leaves, so fall pruning can reduce their cold hardiness. Fall pruning can stimulate succulent growth which may not harden off sufficiently to avoid winter dieback.
Azaleas and other spring-flowering plants would bloom poorly if pruned in fall, as the next year's flower buds would be pruned off. Such plants are best pruned in spring immediately following petals fall. A good reference is Purdue University Cooperative Extension publication: Pruning Ornamental Trees and Shrubs

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