Bitcoin core does not anymore use the notion of priority, where a part of the block was reserved for transactions spending old UTXOs.
Further there is something called a "relayfee", which is simply a policy and default setting that you can check in your node when calling "getinfo". Any transaction with a fee/kb under that one is not propagated by the node. The default value currently 1000 satoshis per kilobyte.
But a miner can still choose to add any (valid) transaction to a block, irregardless of the fees it pays or amount it spends. But rational miners will always try to create blocks that maximize their profits, hence pays them the most in transaction fees.