Well, If one has an infection in the eyes tears can cause burning. When you cry, the tears have salt in them, (Sometimes the salt content is higher if you have not been drinking enough water.) and it causes irritation for your eyes. So that's basically it, or it could be that you cried really hard, and that your eyes burn from the salt, and pressure of your hands rubbing at your eyes.
It is normal to have burning eye when you cry. Because tears are salty, and too much water and salt can lead to eye angioneurotic oedema. So you needn't worry about that. Here are some tips for you to consider. First of all, don't take food with high salt and drink wine. Besides, you can alter with cold and hot compress to your eyes. Finally, don't drink too much water before you go to sleep. After all, keep a good mood and live a happy life will be important.
Well, in my opinion, it may be because eye infection. So you should just be careful about it. Generally speaking, eye infection can just be a serious eye problem. When it occurs, it will just lead to some other symptoms. For example, just like your situation, your eyes will be burning when you cry. And of course, that can be related with your tear system too. Anyway, you just be careful about it. Or eye infection will be more serious.