Be is not always required to mark passive voice. There are other constructions like the one in the example given where the passive is not marked by be (CaGEL p1245):
i Most of the sense verbs
I heard the window broken.
ii Get and have:
She got/had the house painted.
I had my wallet stolen.
iii Like, want, report,fear and order; here the past-participial is an
alternant of a passive to-infinitival:
He’d like/wants them (to be)killed humanely;
The captain was reported(to have been)killed;
They are feared (to have been)abducted;
He ordered it (to be)destroyed.
iv Need and want, as an alternant of the concealed passive:
He needs/wants his hair cut
In order to select the correct complement to the verb like, all we need do is consider what complementation patterns are allowed.
Like can take objects, predicative complements, gerund-participial clauses, to-infinitival clauses, expanded declarative content clauses (that-clauses) and past participial clauses:
Billy likes ice-cream. [object]
Billy like his ice-cream cold. [object + PC]
Billy likes his ice-cream to be served in a cone. [object +
Billy likes eating cold ice-cream. [gerund-participial]
Billy likes me serving him ice-cream. [object + gerund-participial]
Billy likes his ice-cream served in a cone. [object +
Billy likes for his ice-cream to be served in a cone. [to-infinitival]
Billy likes that his ice-cream always comes in a cone.
Given these choices for the complementation of like, we can now choose the correct answer. All three options have multiple possible interpretations depending on context:
a) have finished the report
plain infinitival I will have finished the report.
plain present I have finished the report.
b) the report will be finished
declarative main clause The report will be finished.
bare declarative content clause (no that) He said the report will be finished.
c) the report finished
declarative main clause The report finished.
object + past-participial We got the report finished.
We see that the only one which fits the allowed complementation patterns for like is (C).