Is one required to move the fridge and clean out all the chametz that has gotten stuck underneath? This question assumes that the fridge will not be moved at all during Pesach.
The Radvaz (Teshuvos, chelek aleph: 135) and the Chayey Adam (119:6) explain that the reason we clean cracks and crevices from chametz is because maybe we may come to eat it on Pesach (because it might get into food).
Otherwise, the Gemara Pesachim on 6b says that crumbs are not important at all. The Magen Avraham (S.A. 434) even says that crumbs are "batel" (nullified) by themselves, and do not need your declaration.
The Chazon Ish 122:18 (brings the Gra) and says that visible bits of chametz stuck in between cracks in the floor, do not need cleaning. This is because no one will be suspected of gathering these bits to eat them. The Bes Yosef, Magen Avraham, and Mishnah Berurah (see Tur + S.A. 442) all agree that once chametz pieces are slightly dirty, there is no further reason to actually clean them, since no one will come to eat them. Your "bitul" declaration takes care of everything.
This explains why the Chazon Ish (bringing the Gra again), is strict to clean books for crumbs (but not clean the floor cracks). He explains that books brought to the table with crumbs in the pages may fall out into the food. (BTW Many poskim are lenient even by books for various reasons.)
Shulchan Aruch HaRav 433:19 specifically discusses a case of an actual larger than kezayis piece of good chametz that falls between the floorboards and remains visible. He says that one does not need to actually remove the floorboard and fish it out. The declaration of "bitul" is enough. This is because a person is not going to go there, so there is no risk of coming to eat it. In his Kuntres Acharon, the Rav objects by suggesting that if a valuable object (like jewelry) also fell in between the boards next to the chametz there on Pesach, we would rip out the board to recover the valuable? He answers that you still do not have to go get the bread out, because the chances of dropping a valuable there on Pesach are very slim.
Therefore, one need not move the fridge (or large appliances that are meant to remain stationary) to clean chametz under, or behind it; even if you know there is chametz there. Those who do so anyway (while making sure not to annoy family members or neglect real Halachic issues in the process), are trying to voluntarily serve Hashem with extra fervor for chumras Pesach.
I hope this helps.
As long as there's no reasonable chance of encountering it on Pesach, then no. (This comes up more often as people try to find any chametz behind their fridge, but the same idea -- try sweeping underneath if you can, but you don't have to move the fridge!)
Rabbi Hershel Welcher in Queens has a pre-Pesach shiur on this. "Let's pick one day a year for people to clean out behind their refrigerators -- I don't know, May 15 or something -- but it's not a Pesach requirement! (He later found out that when he said you don't have to clean out behind your fridge, someone eventually misquoted him as saying when cleaning the interior walls of your refrigerator [which you should do], you don't have to clean the rear interior wall.
The mishna in the first chapter of Pesachim talks about what places need to be checked (or for our purposes, cleaned); the concerns are whether chametz is likely there; whether you would really want to keep chametz if it was there; and whether you'd encounter it on Pesach.
Going a bit further than the previous answers, my Rabbi specifically said not to move it within 30 days of Pesach. If it is moved within 30 days, there might be a chiuv bedika.