Do constraints make an optimization problem harder or easier? Assuming that constraints are satisfiable.
In general, does adding constraints make an optimization problem harder since the optimal solution does not necessarily satisfy all constraints?  Or does it make it easier by reducing the search space for a feasible solution?
Perhaps "in general" doesn't make sense at all, in which case, under what conditions does adding constraints make the problem easier and when does it make it harder?

Nobody adds or subtracts constraints from an optimization problem to make it easier or harder. A real-life optimization problem has a well-defined set $S$ of feasible states and a certain object function. Now the set $S$ can be described in different ways, often with too many variables which then are kept under control by constraints. I don't think there is a general philosophy that would dictate in which way to proceed in such circumstances: eliminating superfluous variables can introduce undesirable boundary conditions, or similar, etcetera. 
Maybe you better show us an example where you are really in doubt.
