Final answer:
Cooling the mixture of the given equilibrium reaction will cause the equilibrium to shift to the right, as the system attempts to produce more heat to counteract the cooling.
Step-by-step explanation:
When cooling the mixture for the equilibrium reaction CH₄ + H₂O ↔ CH₃OH + H₂ + heat, the position of equilibrium will shift towards the right. This is because cooling the system removes heat, and according to Le Chatelier's Principle, the system will adjust to oppose the change introduced. In this case, that means the equilibrium will shift to the right where heat is a product, in an attempt to produce more heat and counteract the cooling.