Are the 16, 32 and 64GB models the same weight or does increased memory increase the weight?
As a general answer normally an increase in memory capacity of that order won't have a significant effect on weight. I'm not sure exactly what memory chips Apple use in the iPhone 6 Plus but 16GB is in the realm where a single chip can be used. As an example a TSSOP-56 package is pretty common for FLASH memory and if you read a typical Package Material Disclosure the total maximum weight is 221 milligrams each.
If they added an additional three and you allowed a little extra for solder it would still come in under a gram. So when Apple don't list a seperate weight based on memory capacity they probably either use higher density FLASH or don't consider the weight significant enough to list.
According to Apple the iPhone 6 Plus weights 6.07 ounces (172 grams). There is no mention of weight differences depending on memory size.
My iphones do feel like larger memory is significantly heavier. We have iphone 6 64GB and 128GB. We also just got iphone 7 plus 128GB and 256GB.
I'm surprised why no one reports this difference in weight.(I just spent an hour googling this, so many iphone reviews and discussions but nothing about this issue.. I just don't get it why my wife's phone is lighter..)