I don't know anyone who says "yes" to the above question for small, personal purchases.
Do you need the toilet paper roll of barcodes that's printed every time you go to CVS? How often do you go back to your personal ledger and even up your daily take in coffees from the receipts you're handed?
Paper receipts seem more outmoded because they're paper -- a physical duplicate of a record that exists for any future reference in online banking statements and other, personal financial management tools.
The two responses I got when I tweeted about why we still have paper receipts for every retail transaction were: to avoid employees stealing from registers and for audits. Both valid and both capable of being monitored automatically online when a transaction is completed.
It feels like the hanging chad issue from the 2000 election, though more an annoyance than determining factor in a presidential electoral review. Maybe Square will make this discussion moot. People will realize that receipts can be emailed and records accessed online. But for the sake of everyone who comes home with crumpled receipts in their pockets and for all the trees wasted printing, how are these tiny squares of economic antiquity still necessary?