- How does the Torah view someone who's belching or burping while he's eating a meal? Is a human instinct or animal behavior?
- Is better that he passes gas then burps?
For question #2, obviously, a person cannot choose between the two options, but we will contrast the two options
Believe it or not, there are actually sources that somewhat relate to this question!
The discussion is about burping or passing gas during Davening. Aruch Hashulchan OC 97:2 says the following:
...אמנם החוש מעיד כרבינו הב"י דבגיהוק יכול האדם לאנוס עצמו ולכן נ"ל דלדינא ודאי כן הוא דצריך לאנס את עצמו שהרי אנו רואים כשיושבין במסיבה וצריך לגהק מאנס את עצמו לבלי לגהק מפני שאין זה מדרך ארץ והש"ס והפוסקים שאמרו דלאונס מותר ודאי כן הוא דאם האונס גדול ומוכרח לגהק יניח ידו על פיו ויגהק ורבינו הב"י לא הזכיר זה משום דע"פ רוב יכולין לאנס את עצמו וכן יש להורות:...
He notes that it is not Derech Eretz to burp during a meal. This may mean socially acceptable, or be a Torah-related reason, however, to burp during Davening is certainly considered unacceptable, as we see from this Halacha, where it is prohibited, and compared to yawning (and perhaps spitting).
He seems to think that it is easier for people to hold themselves back from burping than from yawning, and that it is very uncommon for one to burp when he is trying to hold it in. He Paskens that there is no need to move from your spot to burp, and if you cannot contain yourself, simply place your hand over your mouth.
Contrasting this to OC 103 (see SA, Aruch Hashulchan, etc. there), there seems to be an indication that it is more common for a person to be unable to stop themselves from passing gas. However, passing gas (at least during Davening) is much more severe, and according to Shulchan Aruch, a person must move 4 Amos from his spot if he needs to pass gas, as this considered an "embarrassment" towards God.
However, Aruch Hashulchan (103:4), based on Rema, allows such an individual (when Davening in Shul) to stay in his place, so as not to embarrass him in front of others. We see from that source that it would be embarrassing for others to know that he had passed gas, and presumably, this would apply to passing gas after a meal, as the Halacha is discussing a case of Ones.
- The Poskim assume that a status of "Ones" may apply to one who burps as well as one who passes gas during Davening (Kal vachomer to during a meal), thus classifying them as "Human Instict which is sometimes unpreventable". The Aruch Hashulchan makes a comparison to the case of a meal.
- Based on the Halachos and statements of the Poskim in these 2 places, passing gas was considered a more embarrassing act both around others, as well as in regards to Hashem.