The Mishna (Pesachim 49) has a machlokes when to dispose of chameitz when Erev Pesach falls on Shabbos. The Chachamim hold that all chameitz need not be disposed of until Shabbos, but Rabbi Meir disagrees and holds that bi’ur chameitz with the exception of lechem mishne should be done on Friday. What is the crux of the issue? Had you asked me I would have said we are dealing with a simple harchaka derabbanan. Rabbi Meir is concened lest too much chameitz be left on Shabbos without a means of disposal; the Chachamim don’t feel disposing of leftover chameitz poses a concern. One clue that this might be the case is R’ Eliezer bar Tzadok’s view that chulin may be left until Shabbos but terumah must be disposed of on Friday. The reason behind the split seems to be that terumah is harder to dispose of than chulin because it cannot be eaten by non-kohanim. Rabbi Meir perhaps extends the concern for disposal from terumah to all foods, while the Chachamim do not see any danger in either case.
Why is Rabbi Meir more concerned with insuring proper disposal of chameitz on Shabbos than on a regular weekday? My guess would be that since Shabbos has issurim like carrying chameitz to a reshus harabim, burning chameitz, etc. which do not apply on a weekday, proper disposal is more of an issue.
The Ba’al haMaor offers a deeper insight into the dispute. According to Ba’al haMaor, Rabbi Meir holds like Rabbi Yehudah that chameitz must be disposed of by burning, and therefore bi’ur chameitz must take place on Friday. The Chachamim are consistent with their view that chameitz may be disposed of by any means and can therefore be accomplished even on Shabbos.
The Rambam clearly disagrees with this view of the Ba’al haMaor. The Rambam (Chu”M 3:3) paskens like Rabbi Meir that all chameitz should be disposed of on Friday, yet the Rambam also paskens like the Chachamim (Chu”M 3:11) that chameitz may be disposed of using any means.
According to the Rambam, if chameitz can be disposed of using any means, why not simply wait until Shabbos to accomplish that disposal? My approach would answer the question, but resorts to formulating a dispute about a rather vague harchaka derabbanan. R’ Soloveitchik had a different solution… stay tuned (if I have time to write).