Hi. I removed Cairo Dock from Software Sources so that when I ran the Software Updater, it didn't give me the message about "Failed to download repository...". However, I am still using the package and it is installed in Synaptic. I was wondering if removing the listing in Software Sources, but still using the program is a security concern, or does it not matter because Cairo Dock already wasn't supported through Software Sources, hence the message error when running the Updater
you removed the ppa from software sources, not the actual cairo so it remains installed.
tl;dr Don't worry, you did the right thing, no added risks whatsoever.
Cairo dock is no longer maintained by the cairo-dock-team-ppa since
Ubuntu 'Vivid'.
Since then, Cairo-Dock has been added to the standard 'ubuntu-universe' repository. That means that the caito-dock-team PPA can be removed safely. In other words: Your system hasn't been using that PPA for ages so the best thing (you did) is to get rid of it. 
p.s. No it is not a security concern. You are using an up to date version that will keep on receiving updates.
