Pains in the mouth are not uncommon. What is somewhat more uncommon is a not-painful swallow with a painful burp. The case study, “The Wrong Toothpaste and The Painful Burp,” dives into this mystery for a 31-year old man.
To his surprise, there was only slight exacerbation upon swallowing, whereas burping triggered very severe pain with radiation to the ears, lasting approximately 5 seconds.
Researchers used techniques that involved examining an ulcer on his uvula and taking a video of the patient burping (with audio).

Finally, the patient realized that he had accidentally been using a toothpaste containing sodium lauryl sulfate for a few weeks prior to symptom onset. He has not experienced any recurrences upon switching back to his regular toothpaste without sodium lauryl sulfate.
The full citation: Pareek, M. and Bhatt, D. (2017). The Wrong Toothpaste and the Painful Burp. The American Journal of Medicine, 130(1), pp.e19-e20.