I am sorry to hear that your fellow ingested part of a hand warmer.
These packets do contain reduced iron and if enough if that is ingested it can be toxic.
I would be more concerned if he was a small dog. The good news is that he seems fine other than the one vomiting episode and diarrhea.
Ingestions of less than 9 mg per pound of body weight of elemental iron typically result in mild clinical signs but are not likely to result in significant toxicity. I suspect that is likely the amount he ingested given his symptoms so far. 9 - 27.27 mg/pound of body weight of elemental iron can cause mild to moderate signs of toxicity, and eating more than 27.27mgs per pound of body weight can lead to serious poisoning or death. Eating between 45mgs and 113mgs per pound is potentially lethal.
Toxicity can cause direct damage to the lining of his gastrointestinal tract, (leading to vomiting, diarrhea, abdominal pain, lethargy, and gastrointestinal bleeding, sometimes severe), damage to the cells in his body leading to liver damage and difficulty clotting (the liver produces clotting factors) shock, and death. Very rarely we can see strictures forming in the stomach which cause an obstruction 2-8 weeks after eating the iron.
At this point I would recommend that you give an acid reducing medication to protect his gastrointestinal tract as a just in case measure. This will reduce the amount of iron that he will be able to absorb too. You can give either:
1) Pepcid ac (famotidine) at a dose of one 10mg tablet per 20 to 40 pounds of body weight every 12 hours (1 & 1/2 of the 10mg size tablets every 12 hours for a 50 pound dog)
2) Prilosec (omeprazole) at a dose of one 20mg tablet per 40 to 80 pounds of body weight every 24 hours
Either one should be given for at least the next 2 weeks. If you don't notice any other symptoms you can then discontinue the acid reducer after that.
If you notice diarrhea that is bloody, he should be seen by a veterinarian. If he has a tense painful abdomen, refuses to eat, or seems very lethargic he should see a veterinarian.
If the whites of his eyes appear yellow or his gums appear yellow, or his urine is very dark yellow in color he should see his veterinarian to check his liver function.
I also recommend that you feed him small meals of a bland diet for the next couple of days to dilute out the effects. A homemade bland diet consists of 1/3 boiled minced white skinless chicken and 2/3 boiled white rice. Feed this for several days and if all seems well then slowly start mixing in his regular diet, a little more at each meal. It should take about a week to convert him back once you start.
Please let me know if you have any further questions.