How small must an object be in order for it to be subject to the laws of quantum mechanics?
As it is currently understood. All objects are subject to the laws of quantum mechanics.
Observing quantum effects on a macroscopic scale though, is an interesting challenge. To do so we need to eliminate almost all thermal vibration, which destroy the quantum effect, and cool it down to quantum ground state. This requires very cold temperature. For some objects the temperature required is lower than what conventional cryogenic refrigeration can allow.
One such research demonstrates superposition on object as large as 40 micron by cooling it using microwave-frequency mechanical oscillator coupled to a quantum bit.
reference Connell et al 2010 Quantum ground state and single-phononcontrol of a mechanical resonator doi:10.1038/nature08967
In theory, all objects are subject to the laws of quantum mechanics and there is no tension between that and classical physics.
I won't pretend to really understand all of these derivations past waving my hands, but people normally point to Ehrenfest's theorem and the stationary phase of Feynman's path integral being the classical trajectory as the main approaches to the quantum to classical transition.
The latter is also closely related to the Eikonal approximation, though I can't find a good reference at the moment explaining pedagogically how you can use that to show the quantum to classical transition. Further, people like to point to decoherence as the mechanism by which we get classical statistical mechanics out of quantum many body systems.
Sidenote: I could have sworn I read a paper about a millimeter metal shaving being in superposition sometime last year, but I couldn't find it... perhaps it was retracted.