Does purified water corrode copper pipes?
Although it seems pretty clear that removing potentially corrosive compounds from drinking water will result in a net of no effect or possibly decreased corrosion, there is at least one consideration worthy of mention.

While metallic copper has a solubility that is well below most regulatory standards for drinking water (i.e. 2 mg/L in the UK), enough copper can still potentially dissolve to result in a significant loss of copper mass, such that pinholes and destructive oxidative processes can take place.

However, according to the UK's National Physics Laboratory, Good Practice Guide No. 120, "Avoidance of corrosion in plumbing systems", September 2011:  


  "...the rate of corrosion and the release of copper ions slow rapidly in the first few weeks of operation as protective films form on the bore of the copper pipes."  


Further:  


  The surface films (often referred to as a patina) are composed of thin brown or black coloured copper oxide ($\ce{CuO or Cu2O}$) overlaid with a thicker green or turquoise basic copper carbonate layer. These layers, although protective, are not true passive layers (like those which form on stainless steels or aluminium and which are much thinner and invisible to the naked eye). It is when these films on copper do not form properly in the first place or become disrupted later that problems of pitting or blue water can occur.


According to this related document from the UK's "Foundation for Water Research", the low alkalinity and buffering capacity of purified water can result in conditions of slight acidity that can prevent the formation of the desired patina, and recommends the addition of 50 mg/L bicarbonate alkalinity in order to promote the development and maintenance of the protective copper oxide and carbonate layers.  

To reiterate, my answer to the question is a net "No" with an interesting caveat.