Can I use the Bitcoin Core CLI to learn the block height at which a transaction was mined?

For a transaction that's not considered "yours" by the Bitcoin Core wallet, the only way to learn anything about it is through the optional transaction index.

If you have the transaction index enabled (-txindex on the command line or txindex=1 in the configuration file), you can use the getrawtransaction RPC with a txid to find the transaction in the chain. If you add a true after the txid, it'll also report the transaction in decoded format, along with other information such as the number of confirmations. E.g. you'd run:

bitcoin-cli getrawtransaction f4184fc596403b9d638783cf57adfe4c75c605f6356fbc91338530e9831e9e16 true

and get as response:

{ "in_active_chain": true, "txid": "f4184fc596403b9d638783cf57adfe4c75c605f6356fbc91338530e9831e9e16", "hash": "f4184fc596403b9d638783cf57adfe4c75c605f6356fbc91338530e9831e9e16", "version": 1, "size": 275, "vsize": 275, "weight": 1100, "locktime": 0, "vin": ..., "vout": ..., "hex": "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", "blockhash": "00000000d1145790a8694403d4063f323d499e655c83426834d4ce2f8dd4a2ee", "confirmations": 732767, "time": 1231731025, "blocktime": 1231731025
}

If you already know in which block the transaction was confirmed, you can verify it using the same RPC (even without transaction index) by adding the block hash as parameter:

bitcoin-cli getrawtransaction f4184fc596403b9d638783cf57adfe4c75c605f6356fbc91338530e9831e9e16 true 00000000d1145790a8694403d4063f323d499e655c83426834d4ce2f8dd4a2ee

These commands can also be issued in the Bitcoin-Qt debug console.

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