Upgraded to 20.04 and Lost Ethernet Connection

Shortening up this answer:

First, install the dkms driver for the r8168 chipset:

sudo apt install dkms r8168-dkms

Now, unload the current r8169 driver:

sudo rmmod r8169

Blacklist the r8169 driver to keep it from loading in the future:

echo "blacklist r8169" | sudo tee -a /etc/modprobe.d/blacklist.conf

Then load the r8168 driver:

sudo modprobe r8168

Then you can check the status of the driver by running both sudo lshw -C network and dkms status.

Example:

terrance@terrance-ubuntu:~$ sudo lshw -C network
  *-network                 
       description: Ethernet interface
       product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller
       vendor: Realtek Semiconductor Co., Ltd.
       physical id: 0
       bus info: pci@0000:03:00.0
       logical name: eth0
       version: 0c
       serial: 40:8d:5c:4f:12:03
       size: 1Gbit/s
       capacity: 1Gbit/s
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp 10bt 10bt-fd 100bt 100bt-fd 1000bt-fd autonegotiation
       configuration: autonegotiation=on broadcast=yes driver=r8168 driverversion=8.048.00 duplex=full ip=10.0.0.100 latency=0 link=yes multicast=yes port=twisted pair speed=1Gbit/s
       resources: irq:27 ioport:ce00(size=256) memory:fdeff000-fdefffff memory:fddfc000-fddfffff

terrance@terrance-ubuntu:~$ dkms status
nvidia, 460.39, 5.4.0-66-generic, x86_64: installed
nvidia, 460.39, 5.8.0-43-generic, x86_64: installed
nvidia, 460.39, 5.8.0-44-generic, x86_64: installed
r8168, 8.048.00, 5.4.0-66-generic, x86_64: installed
r8168, 8.048.00, 5.8.0-43-generic, x86_64: installed
r8168, 8.048.00, 5.8.0-44-generic, x86_64: installed

Hope this helps!