Let me try to interpret the statements as propositions.
Premise 1: Person B is capable of buying "that" for person A.
Premise 2: Person B cannot afford to buy "that" for person A.
Conclusion: Person B has a responsibility to buy "that" for person A.
I propose that an inability afford something for someone implies an a lack of capability to buy something for someone. Therefore, premise 2 implies:
Premise 2A: It is not the case that person B is capable of buying "that" for person A.
Premise 1 and premise 2A assert contrary positions. (i.e. they are of the form p^~p). From two contradictory propositions, any proposition can be proven (see http://en.wikipedia.org/wiki/Principle_of_explosion).
This works as follows:
Premise 1 is true, and Premise 1 is false.
Premise 1 is true.
Premise one is false.
Premise 1 is true, or the conclusion is true.
The conclusion is true. (by disjunctive syllogism, because premise 1 is false)
There is no fallacy in the argument per se, but at least one of the premises must be false, so the conclusion cannot be inferred. That is the argument is valid but not sound.
Perhaps this exchange seems funny because of the mental incongruity caused by the contradiction between A's and B's first statements.