Would equal masses of gold and copper result in the same volume change if added to water?
"Added to water" is ambiguous.

If the metal objects are shaped and placed such that they float (eg: as a metal boat does, or a light hollow sphere), then they will each displace the same volume of water (the volume of water equivalent to their equal common masses).  So long as they float, they will displace the same amount of water no matter their shape.

If the masses objects are shaped and placed such that they sink (eg: consider a solid sphere or cube), then they will in general displace different volumes.

Bonus.  The second part is slightly more complex than it seems when it comes to calculating how much water they would displace.  If they have no dry internal voids, their respective displaced volumes are simply the mass of each divided by its density.  Others have given the simple calculation.

If they were shaped such that they had dry voids yet were small enough to sink (think of a heavy hollow sphere with overall density greater than water so it sinks, or an overturned metal boat with small air pockets), then the displaced water will be somewhere between the above two calculations (fully floating calc and no-voids calc).

Note that it would be possible to construct the two objects such that they sank and yet displaced the same amount of water, eg: two hollow spheres with the same outer diameter but different internal diameters, both heavier than the same volume of water.  But that would be a very trick question indeed!