Conversion

Fair enough. Lets do it properly and use a union.

union IPAddressConverter {
  uint32_t ipInteger;
  uint8_t ipArray[4];
};

void setup()
{
  // This code will only run once, after each powerup or reset of board
  IPAddressConverter ipAddress;
  ipAddress.ipInteger = Ethernet.localIP();
  char buf[16];
  sprintf(buf, "%d.%d.%d.%d", ipAddress.ipArray[0], ipAddress.ipArray[1], ipAddress.ipArray[2], ipAddress.ipArray[3]);
//twitter.post(buf);
}

void loop()
{
  // This code loops consecutively 
  
}

Oh, I should also mention, you don't need the \0 at the end of sprintf(), it is included for you as part of the " "