I'm wondering if the gate errors on any of the IBM devices are low enough so that we could use some error correcting codes which do not require too many qubits? (such as 5- or 7-qubit codes). Even running programs with on little as 2 logical qubits would be cool.
You might find some examples in this two-year-old question :)
To the best of my knowledge, the most recent work that implements some code on IBMQ's quantum devices is about the repetition code (see the textbook or the paper). If you only want to do the simulation, there should be no problem to take a further step towards more advanced codes. But if you mean real quantum devices, the circuit depth (increased by parity checks and swap operations), and sometimes post-measurement quantum operations would make the implementation quite hard. See this paper for example, which says
Although the fault-tolerant gates offer an impressive improvement in fidelity, the computation as a whole is not below the fault-tolerance threshold because of noise associated with state preparation and measurement on this device.
(I have to admit that it's about fault-tolerant quantum computation and not a very good example.)
Still, like these recent work, you could try to test some QEC codes on IBMQ devices instead of perfectly implementing them.