With some ciders I experience the same burning sensation after a few ciders. I haven't figured out the exact causes yet but I will share a few theories worth testing:
1) Acidity. Some apples are much higher acidity than others. Also some yeasts will produce a higher acid product than others. And as Kingsley also alluded to, carbon dioxide is also acidic so if the cider is highly carbonated, this could be a problem for some people.
2) Sulfites. If your cider is treated with sulfites before and/or after fermentation, be aware that some people are sensitive to it or claim an allergic type reaction to it. This is why commercial ciders and wines will notify whether or not they have used sulfites.
3) Alcohols. The alcohol itself could be causing a problem. And depending on the juice source, if there is haze or starches present, you could have trace amounts other alcohols besides ethanol including methanol or other higher alcohols that your body is telling you it doesn't like.
4) Wild yeast. I believe this is unlikely, however if your source juice was not treated to get rid of any wild organisms that might be present, these beasts could theoretically survive in the finished cider and could potentially wreak havoc to your digestive system. I have my doubts with this one, as I believe alcohol will kill most wild organisms, however like I said this might deserve more testing on a case-by-case basis if the juice is not pasteurized or treated with sulfites.
Theories worth testing. I have not tasted your cider so it is difficult to pinpoint which of these is closer to the right one.