Not when the system is in equilibrium. When a system is in equilibrium, the sum of the moments about any point is zero. It is intuitive to take moments about some point the system might turn about, and this is often convenient and makes forces easier to find, since the forces acting at the point we are taking moments about can be ignored and do not have to be immediately found.
For example, to find the reaction forces at A and B below, we can take moments about A to find the reaction at B and moments about B to find the reaction at A.
Taking moments about A gives
Taking moments about B gives
Now take moments about the endpoints of the rod. Taking moments about the left hand side gives
Taking moments about the right hand side gives
Solving these equations simultaneously:
When the system is not in equilibrium it might be turning about a point – either clockwise or anticlockwise. It could easily be turning clockwise about some point and anticlockwise about some other point – then the point to take moments about DOES matter.