I can't come up with a situation where I would need it.
Elegant systems provide false/0 as a declarative synonym for the imperative fail/0. An example where it is useful is when you manually want to force backtracking for side-effects, like:
?- between(1,3,N), format("line ~w\n", [N]), false.
line 1
line 2
line 3
Instead of false/0, you can also use any goal that fails, for example a bit shorter:
?- between(1,3,N), format("line ~w\n", [N]), 0=1.
line 1
line 2
line 3
Thus, false/0 is not strictly needed but quite nice.
EDIT: I sometimes see beginners who want to state for example "my relation does not hold for the empty list", and then add:
my_relation([]) :- false.
to their code. This is not necessary, and not a good example of using false/0, except for example in failure slices that are programmatically generated. Instead, concentrate on stating the things that hold about your relation. In this case, just leave out the entire clause, and define the relation only for lists that are not empty, i.e., have at least one element:
my_relation([L|Ls]) :- etc.
or, if you are describing other terms in addition to lists as well, use a constraint like:
my_relation(T) :- dif(T, []), etc.
Given only either (or even both) of these two clauses, the query ?- my_relation([]). will automatically fail. It is not necessary to introduce an additional clause which never succeeds for that purpose.