What happened to Dr. Stanley in S01E06?

The scene takes place in the episode titled "Mercy". So I interpreted it as him doing mercy on everyone including himself.

When one of the crew member comes to Dr. Stanley complaining of losing his mind and hallucinating, the doctor brushes it off with mental stress. Then when Goodsir informs him of lead poisoning due to eating canned food, he can connect the dots to that patient.

He doesn't want to die slowly and painfully due to lead poisoning and scurvy. He knows it's coming for him and everyone. Still he maintains a calm demeanor and pretends to not care about it. The calm before the storm. That can be interpreted as denial.

So basically, he sets himself and everyone on fire as a mercy rather than having to die painfully while losing your mind with hallucinations.

There was actually foreshadowing. When Goodsir wants to talk to Stanley, he asks "Where's the fire, Mr. Goodsir?"

First, I can't say I completely agree with your assessment that

...Dr. Stanley is depicted as a largely arrogant and uninterested person without much emotional connection to either the expedition or the rest of the crew.

The British are pretty well known for their emotional reserve (at least in times past), and a medical officer/physician would almost surely display that characteristic. Certainly, he wasn't a very likable character; he was elitist, racist, sexist, and, yes, arrogant. But initially he showed some tenderness towards his patients (like the boy with consumption.)

Having said that, I didn't notice any particular foreshadowing, except that by then, some crewmembers were having significantly frightening hallucinations. Lead neurotoxicity...:

...results in development of irritability, headache, mental dullness and attention difficulty, memory loss, tremor, and hallucinations within weeks of exposure. Symptoms abruptly worsen to paralysis, convulsions, delirium, coma, or death.

While the show didn't show lead toxicity perfectly, it was pretty darn good. The doctor was eating food out of the tins like the other crew members; even Sir John wasn't above tinned food as we see in an early episode when he removes a piece of lead from his mouth during a meal. The doctor was as likely as anyone else to suffer from lead toxicity, and in his case, from delerium.

I believe what we were meant to take home from the "good doctor's" actions is two-fold: 1) how insidious the symptoms of lead poisoning are, and 2) how horrifying the results can be. The doctor appeared to be normal, as did most crew members, but he most certainly was not. He was delusional himself. The treatment for lead toxicity is not to kill everyone with it, and the doctor, if in his right mind, would have had to have known this. Burning to death is an extremely painful way to die; it's not a merciful end for anyone. The knowledge of this pain, the sanctity (then) of the Hippocratic Oath (First, do no harm...), and his setting himself ablaze (arms extended in a Christlike posture^*), the horror of his actions is multiplied, which is appropriate in a horror story.

Finally, this strikes me as the case with the doctor (and some of the other expedition members' behavior as well) to show the audience that the biggest danger to these men at this desperate juncture wasn't the beast hunting them, but their own fellow man.^**

^*_{Delusions of grandeur are a very common type of delusion.}
^**_{Including, unfortunately, the company that provided the tins so carelessly created.}

_{Watching this now, I was certainly horrified!}