Is it obligatory to make up the fast for someone who felt like vomiting and it reached their throat, so they swallowed their saliva fearing that their fast would be invalidated, knowing that they did not intentionally vomit?

If "burping" refers to belching and not vomiting, and the fasting person did not intentionally vomit, then there is no blame upon him for what occurred from belching or for tasting food if he was unable to expel it; because all of that is beyond his will. However, if something reached the throat and it was possible to expel it, but the fasting person swallowed it, then his fast is broken and he must make up that day.