Conversion

Yup, should have checked that.

This would be the modification as suggested by Arrch:

uint32_t ipAddress = Ethernet.localIP();
uint8_t* ipAddr = &ipAddress;
char buf[16];
sprintf(buf, "%d.%d.%d.%d\0", ipAddr[0], ipAddr[1], ipAddr[2], ipAddr[3]);
twitter.post(buf);

However depending on how the ip is returned, you may have an issue with endian-ness in that the IP could come out with the bytes reversed. In which case you would need this:

uint32_t ipAddress = Ethernet.localIP();
uint8_t* ipAddr = &ipAddress;
char buf[16];
sprintf(buf, "%d.%d.%d.%d\0", ipAddr[3], ipAddr[2], ipAddr[1],ipAddr[0]);
twitter.post(buf);