Not with the state of the art today.
Electronic payment systems are certainly a threat to cash. We're seeing more point of sales refuse cash and insist on credit or debit transactions. Cryptocurrencies may ride along on this transition, but they're not driving it. The cost of handling cash is one many businesses would love to get rid of.
For illegal transactions, cash is still more anonymous than cryptocurrencies. A distributed public ledger lends itself to traffic analysis that is much more difficult in the physical world. Cash is also more convenient for fast face-to-face transactions. While it's certain that drug purchases have occurred using cryptocurrency, the vast majority of illegal trades continue to use cash.
Fiat currencies aren't going away either. (And we could argue that many cryptocurrencies are actually fiat currencies, or at least fiduciary money, because they lack use value.) Governments require taxes to be paid denominated in their own currencies, even if some have experimented with accepting payment in Bitcoin. Many governments, like the U.S., require banks to hold accounts at a central bank, denominated in fiat currency., and to hold securities denominated in their account currency. Government benefit programs, employees, and contractors are paid in fiat currency, which constitutes a large percentage of the economy of any modern state.
Until governments permit banking entirely in cryptocurrency terms, cryptocurrency will continue to be a sideline to the "real" economy. Banking provides a lot of needed services, including creation of money through lending, not just payment processing or foreign exchange.
Fiat currency also permits governments to be a lender or buyer of last resort. While this is politically controversial, we should note that even governments that tried to keep on the gold standard abandoned it in times of crisis. Many economists see this ability as a positive, not a flaw. A government run on cryptocurrency might well re-issue fiat currency as soon as spending was needed in a war or financial crisis, just as governments debased metal-backed currency under stress.