The Gemara Avoda Zara (76a) differentiates between kailim used to roast kodshim and those used to roast other issurim. For the former, it is sufficient to kasher with hagalah, because when they had absorbed ta’am it was still heter (heteira bola), while the latter requires libun because when they absorbed ta’am it was already issur (issura bola).
There is a machlokes Rishonim as to how to categorize chametz that becomes absorbed into a kli before Pesach. The Ravad says that since chametz is mutar before Pesach, it too is heteira bola. However, Ramban says that chametz is more chamur than Kodshim since the sheim chametz exists even before Pesach; while kodshim has a sheim heter until it becomes no’sar. Therefore chametz is issura bola.
In general Shulchan Aruch paskens that chametz is issura bola, and therefore kailim used to roast chametz, require libun gamur. However, there are several exceptions to this rule, and these make for differences between the way we kasher for Pesach verses other issurim.
Knives – If one purchase knives from a non-Jew they require libun (Y.D. 121:7). However, Mishna Berurah (O.C. 451:19) explains that one can kasher knives for Pesach with hagalah even though they are occasionally used with chametz directly on the fire (miyut tashmisho), because we have a tziruf of two leniencies. One, m’ikar ha’din we kasher according to rov tashmisho which would only require hagalah. Two, there are opinions that chametz is heteira bola and therefore hagalah is sufficient. Nevertheless, it is considered mitzvah min hamuvchar to either purchase new knives for Pesach or to do libun.
Machavas – A non-kosher frying pan requires libun gamur because we are concerned that the oil might dry up and the tarfus will roast directly against the pan (Y.D. 121:4). However for Pesach, m’ikar hadin it is enough to kasher a frying pan with hagalah because in this case we rely on the opinions that hold that chametz is heteira bola (O.C. 451:11). Rema recommends that a machvas be kashered with libun kal. This is provided that the pan is not merely sprayed with a thin layer of oil, but rather a significant amount of oil is used1.
For example, a kosher chametzdik brazier (tilt skillet) m’ikar hadin can be kashered for Pesach with hagalah, since it is typically used with plenty of liquid on the bottom. However, since occasionally is can be used to fry pancakes with only a thin layer of oil, it is best to kasher with libun kal, provided it will not be ruined.
Milui v’irui gimel yamim– A tank in which cold chametz, milk or non-kosher wine was kavush can be kashered by filling it with water, letting it sit 24 hours and then emptying, and repeating this process three times. Shach (Y.D. 135:33) says that we allow milui v’irui for chametz because we rely on the opinion that it is heteira bola. Although stam yayin is issur, Chazal made an exception, but all other issurim that are kavush, require hagalah. Another leniency for kavush of chametz is that even if the tank is cheres or glass it can be kashered. This is provided we know that the only concern is kevisha, and there is no chance that hot chametz was poured in.
Chanan – Since chametz before Pesach is heter, we do not say chaticha naseh neveila on chametz before Pesach. For example, if a sauce containing 3% flour was added to a soup before Pesach, the soup pot would not need to be kashered, provided that the amount of flour in the soup was batel b’shishim, even though the sauce was not batel shishim in the soup. Of course the pot would need to be cleaned out perfectly.
Kashering ben yomo – Shulchan Aruch (O.C. 452:1) permits kashering kailim for Pesach even when they are still ben yomo so long as the kashering is performed before the fifth hour, erev Pesach. The Magen Avrohom (452:1) explains that in reality this is a nat bar nat bar nat bar nat (chametz → into kli, kli →into water, water →back into kli, kli →into kosher for Pesach product). However, Biur Halacha (452:s.v. she’ain) questions why we are not machmir for opinions that hold that chametz is issura bola and require aino ben yomo2. Therefore, lichatchila we only kasher for Pesach when kailim are aino ben yomo unless we are pogem or the bliya is batel b’shishim.
1 Mishna Berurah 451:65
2 See Chazon Ish (Hilchos Pesach 119:14) for an answer to question of Biur Halacha.