The reason why /boot becomes full is because there are too many Linux kernels installed. The solution is thus to remove some of them. Unfortunately, apt-get dist-upgrade might not work in this situation, so we need to use dpkg directly.

First, run this command to check what packages you are about to remove:

$ sudo dpkg -l 'linux-*' | sed '/^ii/!d;/'"$(uname -r | sed "s/\(.*\)-\([^0-9]\+\)/\1/")"'/d;s/^[^ ]* [^ ]* \([^ ]*\).*/\1/;/[0-9]/!d'

Make sure you are not about to remove the kernel that you are using now. You can get the version of the current kernel like so:

$ uname -r

When you are ready (meaning you did the necessary backups, etc.), remove the extra kernels using this command:

$ sudo dpkg -l 'linux-*' | sed '/^ii/!d;/'"$(uname -r | sed "s/\(.*\)-\([^0-9]\+\)/\1/")"'/d;s/^[^ ]* [^ ]* \([^ ]*\).*/\1/;/[0-9]/!d' | sudo xargs dpkg --remove