Better options for induction coils than water-cooled copper
Commercial Induction Heating Equipment is all about the economics of the equipment and the process of heating. 

There are commercial units using super conductors Seen in this post  and in This Post 

Also, many uses of Induction Heaters are one of a kind for a specific product. Fabricating exotic induction coils for a one time sale might not make sense.  

Induction Heating metals of low conductivity (aluminum and copper) would be candidates for using super conduction coils.  Induction heating of copper components using copper coils results in efficiency less than 50%. 

Heating ferrous objects (below the curie temperature) is probably not a good candidate for exotic induction coils since the efficiency could be quite high (85% to 90% efficient).  However in multi-megawatt systems running continuously could save lots of power and $. 

The efficiency gain of using silver rather than copper would likely still require water cooling. 

Flux concentrators ,  see Fluxtrol also help with efficiency of the heating process.  (not an endorsment for fluxtrol).  

Flux Concentrator taken from Induction Presentation 


The wall thickness of the copper tubing can be made thinner, so as to increase cooling.  But, decreasing wall thickness begins to harm the efficiency (skin depth limitation).