Based on John Isaacks' question, "If you view the Earth from far enough away can you observe its past?" and the responses, it appears that we could use mirrors to see into the past. Using Vintage's example, a single mirror on the moon would only allow an observer to see 0.1μs into the past. Would it therefore be theoretically possible to align a massive series of mirrors that sent a single beam of light back and forth over a larger span of time (e.g., months, years, decades)? If so, would it be possible to send messages to ourselves in the past via such an arrangement? Or would we always be looking into the past and only able to send messages to the future?
Not with currently known physics.
Sending a signal to yourself would require sending a faster than light signal to something that could retransmit it to you and that was itself moving almost at the speed of light. This works because of the way that we cut spacetime into $E^3$ slices. The transmitter moving at close to the speed of light would have its slices tilted relative to ours in such a manner that a superluminal particle could move into its future while moving into our past. The book "It's About Time: Understanding Einstein's Relativity" by N. David Mermin has the clearest exposition of this that I've seen.
Light, by definition, moves at the speed of light. So light could not be used.
We would require not only particles with superluminal velocities but also the ability to control those particals to send a signal with them.
You could not look into the future, all you could do is look at photons that have traveled a long distance, and reach their destination some time after they have been fired, not before they have been fired.
Bouncing light off a distant mirror allows you to "see into the past" because it takes the light a while to get to the mirror and back. You can achieve the same goal much more efficiently by just recording the event or information, and then just letting the recording go into the future the usual way.