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. 

"...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. 

"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.