The last machine on my home network still running Debian Squeeze was my PC Engines ALIX2 based router. The ALIX2 doesn't have any video output though it does send all bios output to the serial port. I also configured grub, the kernel and getty to all use the serial console for input/output by following the instructions in the Remote Serial Console HOWTO. While I did make sure I had a USB-to-serial adapter handy, I didn't require it and managed to complete the upgrade via SSH.
I followed the Debian Wheezy upgrade instructions without incident. The only tricky item was the kernel:
Debian's 686 kernel configuration has been replaced by the 686-pae configuration, which uses PAE (“Physical Address Extension”). If your computer is currently running the 686 configuration but does not have PAE, you will need to switch to the 486 configuration instead.
You can check for PAE support by checking /proc/cpuinfo. The Debian instructions provide a simple grep command you can copy and paste to look for the correct flag. The ALIX2 doesn't support PAE so I removed the installed kernel and replaced it with kernel-image-486 as instructed. The grub config was automatically updated though I did double check. You can run update-grub to update grub if required.
After a reboot the system came up with the new kernel running Debian Wheezy. Debian Wheezy is still a couple of days away from official release however I have been running it for weeks on a laptop, a kvm host and a bunch of virtual machines without any issues.