According to special relativity, has the future already happened?
This is an idea known as the block universe. The example of a photon is a bit extreme, since photons have no rest frames, but the idea is the same: to a fast-moving observer, our future might lie in their past, suggesting that our future already exists.
However, these are purely philosophical notions. In order to see what physics has to say about it, we need to translate "the future already exists", which is a bunch of words, into an experiment that can actually be performed.
One such experiment would be to attempt to receive signals from the future, such as the result of a coin flip. However, relativistic causality forbids this from happening, and also rules out any and all similar experiments. Physics really has nothing to say about this question.
You might complain, why would we have to experimentally test this? Isn't the future already existing directly baked into the math of special relativity, which we know is true?
It is, but you have to be careful not to mix up features of the mathematics of a theory with the theory's predictions. Showing that the predictions are correct does not show that the mathematical structures used to make the predictions are ontologically real. For example, quantum mechanics has a huge variety of flavors, all of which have different ontologies (wavefunction collapse, many worlds, pilot waves) but identical predictions.
Physics can't distinguish between them. Deciding which interpretation of quantum mechanics is right, or whether Lorentz invariance says anything about free will, is a job for the philosophers.
There's a famous story about Einstein, as a teenager, wondering what it would be like to ride on a light wave, and this line of inquiry developing into special relativity over the next decade. The story as usually told skips over the interesting part: it turns out, nontrivially, that it doesn't make any damn sense to imagine you're riding on a light wave. From the perspective of "an observer" riding alone a light wave, you have regions of static electric field without any source, which doesn't obey Maxwell's equations. The only way it's possible for Maxwell's electrodynamics to work for all observers is if none of those observers can ever "catch up" to an oscillating field, which moves at finite speed.
The worldline of a photon in special relativity is made up of a series of causally connected events: emission, transmission, perhaps reflection, absorption. The absorption of a photon is an event in the future light cone of its emission. And the light cones are not deformed by the Lorentz transformation. There's no pathological coordinate system you can choose to move an event from my "future" to my "elsewhere."
For the answer, instead of a photon, imagine a fast spaceship moving near light speed. According to the twin paradox, the astronaut could reach far stars and turn back to Earth within his lifetime. However, in the meanwhile, thousand years may have been passed on Earth.
In this example, the astronaut has reached our future, due to the simple fact that he moved near light speed, and due to time dilation. However, time dilation never transports you into the past of somebody, that would entail causality problems.
It is important that time is advancing differently for each particle, and the twin paradox is working also at small scale. For example, a busdriver is aging slower than a prisoner, his clock is going slower. Even if the time difference is astronomically small, the difference is there. Each particle is following its own proper time, and the right hand of a righthanded tennisplayer will see the future of the left hand.
Let us return now to the photon. We might observe a photon from the early universe, which had been emitted billions of lightyears ago, the photon has reached the future of the particles of the early universe of which we are made today. Each particle is following its own worldline and by this the pace of its own clock. Time dilation depends on velocity ($\gamma (v)$) and on gravity.