I use an ethernet shield and SD together on a Due. Both work fine.
The secret is D10. Do not set it to OUTPUT. It will lock up the w5100. Use the weak pullup on D10 to disable the w5100. Here is the setup I use that works.
void setup()
{
Serial.begin(115200);
pinMode(4,OUTPUT);
digitalWrite(4,HIGH);
// disable w5100 SPI while starting SD
digitalWrite(10, HIGH);
Serial.print(F("Starting Sd.."));
if(!SD.begin(4)) {
Serial.println(F("failed"));
}
else {
Serial.println(F("ok"));
}
Ethernet.begin(mac, ip, gateway, gateway, subnet);
Serial.println(Ethernet.localIP());
delay(2000);
server.begin();
Serial.println(F("Ready"));
}
edit: Here is my Due/ethernet shield/SD web server in action.