Monero block rewards will never drop to zero. Block rewards will gradually drop until tail emission commences at the end of May 2022. At this point, rewards will be fixed at 0.6 XMR per block.
Miners need an incentive to mine. Because of the dynamic blocksize, competition between miners will cause fees to decrease. If mining is not profitable due to a high cost and low reward, miners lose their incentive and will stop mining, reducing the security of the network.
Tail emission ensures that a dynamic block size and fee market can develop.