I have updated the r8168 driver:http://www.icjacobsen.com/dld/oem.tgz
It is now a backport to ESXi of the latest r8168.011.00 driver and on top of that has extra protection built in to avoid package dropping.
For anyone who has observed problems with the old r8168 driver, please test it.
You can upgrade by using a ubuntu live cd and a usb stick:
1. Copy the new oem.tgz file to your machine.
2. Make custom changes to it if needed.
(or rip out the r8168.o driver from the above file and copy it into you previously made custom oem.tgz file).
3. Copy the oem.tgz file to a usb stick.
4. Start your ESXi machine using the ubuntu live CD (boot into the live cd) and plug in the usb stick.
5. Copy new oem.tgz file from the usb stick to the partition that holds the existing oem.tgz file (I think it is partition 3, it is located in the root directory of the partition).
6. Shut down the server.
7. Start it up again normally and test.