If I increase my debian size here:
https://manager.linode.com/linodes/disk/mylinodeIDhere

...will it continue to run as before, after boot? Or does the increase in size affect the already-used data?

Thanks

Yes, it should continue to work as before.

But if the disk contains a filesystem, you might want to resize the filesystem as well, because otherwise there would be no point in making the disk bigger. This (filesystem resizing) is where you have to be careful. Some filesystems (ext4) need to be unmounted to be resized. Others (btrfs) need to be mounted to be resized.

Well, I went from 2G to 6G, and the one program we had on it has starting working again. It had stopped working when the disk filled up. So I'm guessing I don't need to look at the file system, which is good because I don't know how.