Pls don't worries, the problem you said is common, and I often have that sense of that like you. Long time ago, I have just asked my doctor about the reasons why I feel like that. Now I know the reasons. As you know, there are some salts naturally in our tears, and your skin is dry that will result in some hurts( they are so small that you can not see them). And when the salts come into the dry skin on your face, the salts will make your hurts painful, and the pain just like your skin is burning. Of course, there are some ways for you to avoid the problem. For example, before you go to sleep and after washing your face, you can use some cream to make your face feel wet and it will not be dry. Then also, you can try your best to avoid crying, especially in winter. Thanks.
There is no need to worry, it's normal. The tears are hot if you cry with emotions.
So you may feel the hot. And tears contain some potassium, which makes our skin feel burning. It's just a kind of chemical reaction.
Imagine the boiled sea water with skin. It may be a little bit exaggerating, but the theory is the same.
Sometimes tears can irritate people's eyes and skin under your eyes since some salt in the tear. But if it burns your skin under eyes, you may have sensitive skin and it is not normal.
First you can use some cream on your skin for a while to calm your skin down and then go to see an doctor and see what he ask you to do.