Does memory size increase the weight of the iPhone 6 Plus? 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.