For straightening hair with a flat iron:
This source provided by bobthejoe states that temporarily straightening hair involves the breaking of hydrogen bonds between keratin molecules within hair.
One of the reasons hair is curly is because of hydrogen bonds between the proteins (keratin) that make up your hair [...]. Another way to alter hydrogen bonds is with heat; electric straighteners work because of this principle. The platters on electric straighteners are flat so that when your hair cools it takes the shape of "flat" as the hydrogen bonds reform. The same thing occurs with curling irons, but since the heating element is circular, the hair stays curled as it cools. The effect of heat, however, is temporary, over time the hydrogen bonds eventually return to their original form and the hairs goes back to the way they were. This rearrangement happens because moisture in the air hydrates the proteins.
This source provides a similar explanation.
Running a flat iron through blown-dry hair breaks the hair’s hydrogen bonds. It flattens hair until water or water vapor (from humidity, drizzle or the shower) penetrates the hair fiber and allows the hydrogen bonds to revert to their natural positions.
Thus, the straightening of hair with an flat iron does not involve the breakage of disulfide bonds as I had originally hypothesized.
For perming hair:
This article suggests that perming involves the chemical treatment of hair with reducing agents to break the present disulfide bonds between cysteine amino acids, followed by the reformation of the disulfide bonds in the desired curled shape.
In a standard “cold” perm, hair is put into curlers and the reducing agent ammonium thioglycolate is added. The disulfide bonds break and keratin molecules are now free to move around and adjust to the shape of the curl. Then a “neutralizer,” such as hydrogen peroxide, is added to reverse the effect of the reducing agent. New disulfide bonds form so the keratin molecules are locked into the shape of the curls.
This Wikipedia article also suggests that the chemical straightening of hair is obtained with the same process.