On first read everyone focuses on the fact that Harry needed to die and on Snape’s sympathy for that, but on reread I can’t help but pay attention to Dumbledore.
This conversation started with Dumbledore calling Snape to his office, but when Snape got there, Dumbledore didn’t speak to him right away. Instead he paced around talking to himself.
“Harry must not know, not until the last moment, not until it is necessary, otherwise how could he have the strength to do what must be done?”
“But what must he do?”
“That is between Harry and me. Now listen closely, Severus. There will come a time — after my death — do not argue, do not interrupt! There will come a time when Lord Voldemort will seem to fear for the life of his snake.”
Come on, you were the one who called Snape over to brief him, why talk about something that had nothing to do with him? Dumbledore was too anxious before saying the plan that sent Harry to die. He rehearsed it to himself to the point of ignoring Snape.
Along with how he snapped back at Snape's question and didn’t allow Snape to interrupt, you can see just how unstable Dumbledore’s emotions were.
“Tell him what?”
Dumbledore took a deep breath and closed his eyes.
...
Dumbledore opened his eyes. Snape looked horrified.
While explaining the whole plan, Dumbledore kept his eyes closed, shutting down his emotions. He couldn’t guarantee that with open eyes he wouldn't feel ashamed under Snape’s looks, or that he’d get through the plan without leaking how much pain it caused him.
Now you tell me you have been raising him like a pig for slaughter —”
“But this is touching, Severus,” said Dumbledore seriously. “Have you grown to care for the boy, after all?”
When Snape questioned Dumbledore, Dumbledore turned it back on Snape and asked about his feelings for Harry. This was partly probing Snape’s own personal growth, and partly deflecting, because Dumbledore had no face to say he loved Harry right after saying Harry had to die, neither cound he just lie and say he never loved Harry and just treated him like a pig for slaughter.
Finally, the tears after he saw Snape’s Patronus. Maybe they weren't just tears moved by Snape, but also from grief for Harry that Dumbledore had been holding back.
Dumbledore knew Harry would likely survive, and that only by destroying the piece of soul inside Harry could Harry kill Voldemort permanently and go on to enjoy his life. Yet the possibility of Harry actually dying, and the plan of “sending Harry to die” itself, still caused Dumbledore enormous pain. Yet he managed to hold back and carry out the plan after all. I think that's some delicate writing.