When all coins are mine, the block reward become zero but the tax included on that block will be the reward of the miner.  It means, the mining activity will continue since there are still transaction fee to be mined.  Satoshi created this kind of system to limit the inflation of Bitcoin and at the same time encourage the use of Bitcoin in order for the network to continue.

So I think if the Bitcoin adoption (Bitcoin usage) is very successful, miners will have the motivation to keep on mining because of the huge transaction fees to be mine.