Explain how would the energy conservation law fail if electric current is assumed as a vector quantity?

Perhaps you need to double check the question? Or provide
greater context?

The fact of the matter is that electric
current IS a vector quantity, and the energy conservation law does NOT fail. So some
added detail is needed to answer your question.

Current is
a vector quantity because it is made up of moving charges; the charges are moving in
some direction; direction + quantity = vector
quantity.

Consider the energy conservation law
written in the form of Poynting's theorem:

du/dt =
-∇·S - J ·
E ( · is the dot product
)

Here, J is the current
density vector; note that I, the current vector, is
qJ. So I, even as a vector,
satisfies the conservation of energy. Note that current is always a vector, even if it
is all flowing in only one direction.