For me, the saddest scene in a TV series is Billy's death in Ally MacBeal. 

I consider this one of the saddest scenes in a TV series because prior to the character Billy's death, I believed that he and the lead character, Ally MacBeal have always been in love with each other. However, because of circumstances (separate colleges, Billy getting married, then going through a mid-life crisis where he becomes a class-A jerk) they never really got together. 

Then, in his final scene, as he was giving a closing argument in court, which turned into a profound declaration of love (finally) for Ally, out of the blue, he just collapsed and died in her arms. But to make matters worse, Ally had to meet with Billy's ex-wife and her friend, Georgia, where repeats Billy's love declaration, but this time, telling it as if it was meant for Georgia, and not her. 

So, imagine being in love with one person, reconciling with yourself you will never be together, and then finally hearing how the other person feels the same way about you, only to have it taken away in the blink of an eye. And, as if that wan't enough to break your heart, you have to crush it further by giving away your final moments with the person to someone else. I just thought this was one of the most heartbreaking scenes in television.