Can PoE cause interference with neighboring non-PoE Ethernet cables?
No, not a bit. Twisted pair cables are HIGHLY resistant to interference pickup. Furthermore, DC does not cause interference, since it is basically an unchanging current, so there's no change in magnetic fields from it to cause interference (other than when turned on, and off - and the twisted pair cables will reject that interference by design, anyway.)

I have many bundles of cables including POE cables in my work role. Interference is not a factor.