What is Gregor's role in the family, why does he have so much responsibility, and how does this role change?

As gpane's very clear and thorough answer shows, Gregor
brings in the only income in the single-income family, supporting his mother, father and
sister.

The pressure on him suggests the family's lack of
resources and, perhaps, its lack of resourcefulness. Should we see the Samsa family as
lacking imagination or should we see them as a family that would like to shirk work?
Should we see them as a family adhering to social forms and so unwilling to entertain
the notion that anyone but the lone son should be a part of the labor
force? 

There is an implication that the family is
concerned with perceived social forms in addition to its practical
concerns. 

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"...what mainly prevented the family from moving
was their complete hopelessness and the thought that they had been struck by a
misfortune as none of their relatives and acquaintances had ever been
hit."

Gregor's disturbing
transformation is socially awkward, of course, but its greatest impact is a negative
shift in the family's financial situation. This fact is arguably the most prominent
element of the story and informs Gregor's emotional situation to a considerable
degree. 

The family's willingness to rely on Gregor - then
turn on him - suggests a willingness to also exploit him. However we characterize the
family's mentality, the fact remains that when we meet them they rely on an implied
assertion - only Gregor should work. 

The firm he works for
appears as demanding as the family. 

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"What a fate: to be condemned to work for a firm
where the slightest negligence at once gave rise to the gravest
suspicion!"

Gregor's value as
a person, it seems, is derived entirely from his willingness to submit to toil. Taken
for granted as a wage-earner and as an employee, Gregor is just a "bug" in the system,
as it were, identified with a function and not attributed any qualities of humanity.
While we may certainly want to be more sympathetic to the family in reading the story,
there is ample evidence to suggest that the commercial and social expectations of the
family lack a sense of humanity and instead focus on the
perfunctory and the superficial.