Yes, sooner or later... But maybe not to crypto currency yet, maybe some other cashless methods. Have you ever seen people paying by cash in StarTrek, that's the answer to your subject question 
Yep, absolutely, surely cash will be be history, already cash is being reduced in few countries and digital usage is being promoted heavily.
Special rebates are being introduced in my country to transact digitally. Also new ways are being implemented where fund transfer has no fees provoking users to go by that path. Cryptocurrency will not make our world cashless, it will be the legal tender itself and government which will do so.
Cash will never become extinct. It's been here for centuries and it will stay here as always. The changing thing is that some digital assets will be the new friends of cash, that's all. No need to discuss weird thing about the removal of cash.