In The Road, what are three possible interpretations of the father's act of firing the flare into the sky?

Great question! Of course, when we think about the
symbolic meanings of acts like the one you referred to, it is often not spelled out
directly in the text, but there is enough evidence to infer the meaning. Certainly one
meaning is given for us when the father shows the flare gun to his sun and says they
will fire it:

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We can shoot it.

For
real?

Sure.

In the
dark?

Yes. In the dark.

It
could be like a
celebration.

The boy himself
labels the act of firing the gun with a celebration, presumably to celebrate their safe
arrival at the coast.

The actual description of the firing
of the flare gun leads me to think that it is also symbolic of
hope:

The
flare arced up into the murk with a long whoosh and broke somewhere out over the water
in a clouded light and hung there. The hot tendrils of magnesium drifted slowly down the
dark and the pale foreshore tide started in the glare and slowly
faded.

The flare managed to
penetrate the "murk" of the sky, symbolic of the state of the world, and still gives
light. This to me is symbolic of the hope that the boy has that there is a chance of
survival and there are some "good guys" out there to help
him.

Lastly, the firing of the flare gun could be said to
represent a memory of what times were like before and the limitations of humanity.
Interestingly, after seeing the flare, the boy says:

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They couldn't see it very far, could they,
Papa?

This could be read as a
recognition of the limitations of humanity and how they created the mess that the boy
and the father are trying to survive in.