There are threads about this topic already:
- Esp8266 / esp32 with Arduinos ethernet.h library problem
- [Auto solved...] ESP32, W5500 and Ethernet.h - no DHCP, no visibility in network
Basically you have to use a fixed version of the Ethernet library that works for the ESP32 (open issue). You can use my version of the library (GitHub - maxgerhardt/Ethernet: Ethernet Library for Arduino) with the provided examples and also OTA.
Firebase is available as a library (GitHub - mobizt/Firebase-ESP32: 🔥 Firebase RTDB Arduino Library for ESP32. The complete, fast, secured and reliable Firebase Arduino client library that supports CRUD (create, read, update, delete) and Stream operations.) but hardcoded to WiFi in a few places it seems. You can create a fork and replace WiFi.h with Ethernet.h functions.