While it is accepted that the mitzva of biur chametz can be accomplished in different ways (see e.g. Shulchan Aruch O.C. 445:1 and Mishna Berurah 445:1-2, cited here), there is also an accepted practice to specifically accomplish the biur by burning the chametz (see Ramo ad loc). This may be based on the view of Rabbi Yehuda in the Mishna in Pesachim 21a. The Kaf Hachayim (445:11), suggests that specifically burning the chametz symbolizes burning of the evil inclination.
According to Rabbenu Tam (Tosfoth Pesachim 12b s.v. eimatay) even Rabbi Yehuda holds that before midday of erev pesach one can dispose of chametz in any way. Nonetheless, the Tur (OC 445:1), basing himself on the opinion of Rashi (and arguing with his father the Rosh [Pesachim 2:3]), suggests that there is a mitzvah to burn the chametz even before midday.
The Vilna Gaon (ad loc) suggests that even according to the chachamim who argue with Rabbi Yehuda, one should ideally burn the chametz. However, the reason it is not obligatory is because, even according to Rabbi Yehuda the chametz would need to be burned after its prohibited time. Since we do our biur before that time, it can really be done any way.
According to those opinions that there is value in specifically burning chametz even before the time it becomes prohibited, is this also true when the person will still have other chametz that he will not dispose of by fire? Or is the idea of biur through fire only of value if the act of burning results in there no longer being any chametz in the person's possession? For example, this year when Erev Pesach falls out on Shabbath and most have the practice to continue to eat chametz until Shabbath morning, is there any value in doing a token burning of some of one's chametz - e.g. of that found during the search Thursday night?