Currently, transactions cost at least 0.002 XMR per kB.
If a miner includes a transaction with a lower fee, will his block be rejected by the network?
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The block will not be rejected if it's included in a block mined by another miner who accepted it. See src/cryptonote_core/tx_pool.cpp, line 137 (or just after FEE_PER_KB).
This implies a miner is free to mine transactions with a fee lower than this, and any block found with such transactions will be accepted.