It has happened to me twice in my airline life.
Once when I was flying the CRJ, long ago. We arrive at the terminal and all but one passenger got off. The FA came up and told me, "This guy back here doesn't look very good." I turned in my seat and I could tell from there he was dead, but we called for the CFR folks anyway. I walked back to get a closer look and told the flight attendant he was doing a whole lot better now because he was dead. CFR showed up and carried him off so he could be pronounced DOA at the hospital because they knew we had to go. That was nice of them to do. Of course the FA freaked out and had to go home. We got a new FA and finished the day. The FO with me was another Army guy, and I'm sure it looked rather callous, but we sat in the cockpit and ate our sandwiches while the CFR people were removing the dead passenger. We certainly didn't want to miss out on our quick-turn lunch time.
Second time I was flying Narita to Kona in the 747 Classic. About four hours into it, a FA came up and said she was pretty sure an old lady had died. We continued to Kona, and the Coroner showed up at the same time the Paramedics did, which saved a lot of time.
Now, for the military side of my life, that's another story which will remain quiet for the time being.