Order-by-disorder, whereby fluctuations lift an accidental classical ground state degeneracy to stabilize a subset of ordered states, is a recurrent and prominent theme in the field of frustrated magnetism where magnetic moments, or spins, are subject to competing spin-spin interactions. Thus far, such a phenomenon has been discussed in systems where the quantum ground state is not a "classical"product state. In such a case, both thermal and quantum fluctuations act to lift the accidental classical degeneracy, begging the question of whether one mechanism of order-by-disorder is possible without the other. In this paper, we present results exposing an uncharted route to order-by-disorder, one without quantum zero-point fluctuations, in the ferromagnetic pyrochlore Heisenberg system with the Dzyaloshinskii-Moriya (DM) interaction as the leading perturbation. We prove that any collinear ferromagnetic state is an exact eigenstate even in the presence of the anisotropic DM interaction, while thermal fluctuations give rise to a preference in the magnetization direction. Using linear spin wave theory, we find that the anisotropy appears at lowest order as a subleading term in the low-temperature expansion of the free energy, proportional to T7/2. Our results thus show that the phenomenon of thermal order-by-disorder can in principle occur even in the absence of quantum zero-point fluctuations driving quantum order-by-disorder - this being so in particular when the accidentally degenerate ground state of the classical model turns out to be an exact eigenstate of the quantum version of the model. However, and in addition, we find that when the DM interaction is large, the fully polarized ferromagnetic ground state becomes unstable for a spin-12 system within the framework of nonlinear spin wave theory, a result that is presumably closely related to the recent report of a quantum spin liquid in this spin-12 model at D/J≈2 in Ref. [Lozano-Gómez, Proc. Natl. Acad. Sci. USA 121, e2403487121 (2024)0027-842410.1073/pnas.2403487121].