Let me try to explain with a different situation:
Suppose you're standing in a long queue to buy tickets to your favourite movie. Now, the guy at the counter goes away to have couple of beers despite the long queue. What would happen? The crowd standing in the queue goes crazy. They'll shout at the Theatre owner there and go home shout again in front of the Internet and might never go back to that Theatre again. If the guys at the counters failed to provide tickets at all the Theatres, no one would go to theatres to watch a movie.
Here the Theatre can be a crypto wallet like Bitcoin, Monero etc and the guys at the counters can be the miners except that the guy at the counters are entitled to do that job whereas miners/public are not. This might not be the best example but slightly resembles the situation.
So technically, all the transaction requests made by people will not be verified but will remain in the pending queue and hence wont be part of the blockchain. If the pending transaction requests queue keeps on increasing with no actions being taken then one fine day all users will lose faith on the system and the consequences will be terrible for those who invested a lot in it.
In my opinion, public miners will never give up mining as long as people are trading crypto currencies because lesser the miners more the gas money. Even if it happens, crypto wallets might arrange their own infrastructure, not to accept anymore trading but to mine the pending transactions and settle them. Although very much unlikely but can be a possibility.