Are there any disadvantages to running a stainless steel bike chain? We are currently using KMC Z51RB rust resistant chains on our bikes, a competitor has started to use stainless steel chains.  Does anyone know the advantages and disadvantages of using stainless steel and should we be thinking of changing to SS too?

Many thanks.
Your question doesn't clearly state your criteria in choosing a chain except that your competitors are using stainless steel and you use an alternative rust resistant chain, so I'll answer generally on the assumption that corrosion resistance is the main criteria.  

As far as I know there are no bicycle chains that are completely stainless steel.
You will note on the Connex Wipperman stainless steel chains that only the inner links are stainless steel, so you won't get complete rust or corrosion resistance by using them. You will still need to lubricate and clean the chain to avoid corrosion and increase the chain's life.
The stainless steel in those chains may give you longer wear, with a slight weight penalty as seen in this review.

You will have noticed from the Connex page that I linked to before, that there are a number of other corrosion resistance mechanisms including nickel and brass coatings. Other manufacturers also have a variety of means of corrosion resistance.

The main disadvantage of stainless steel would probably be the expense and possibly a slight weight penalty, although the review I linked to suggests the longer life will offset the expense for their purposes.

Overall though, there is no zero maintenance bicycle chain that I know of that is completely rust proof and maintenance free. Only a belt-drive bicycle transmission will not require lubrication and will not rust and it is claimed they will not stretch. Yet there are other issues with belt-drives, which you can research separately to this question if you are interested.