In the doctrine of no arising does the past and future still exist?
Dogen is saying that from Buddhist perspective, a dharma does not "become" another dharma. Each dharma is a "presence" at its time, static in its own quality, followed by another "presence-quality" and so on.

So wood is wood and ash is ash, it's not the same "thing" changing while moving from point A in time to point B.

To this I can add that the temporal succession of dharmas is similar to a set of real numbers - each individual dharma is a generalization, and we can infinitely "zoom in" on a particular moment between "wood" and "ash" to see a some-level-of-burning-in-progress-dharma.

Wood and ash are connected as cause and effect. Ash exists due to wood (among other things).

Similarly, it's not like a living person dies. They exist in succession, first life, then death. Life and death are connected like cause and effect. Death exists due to life (among other things). Therefore, there is no perishing of the living.

Correspondingly, there is no arising. It's not like the newborn was dead before it was born. Nor is it like the mother turned into the newborn, nor mother's egg, nor the sperm. Instead, these dharmas exist in causal succession.

Now, if you think about it, it's not like any given dharma has exactly one dharma as its predecessor - not at all. Instead, most if not all dharmas have many, many "causing" or "conditioning" dharmas. Which one we consider "the main" cause is entirely up to us.

In this sense, someone who lives now is a "dharma" that has quadrillions of other dharmas as its "inputs" - going up the infinitely forking chains of physical ancestry, informational ancestry, circumstantial conditionality and who knows what else. 

This is why there is no arising, because it's an infinitely spanning network of static  non-discrete dharmas causing and conditioning each other. The whole thing is like a multidimensional vector space, without one global time dimension.



So, to answer your question, 

Past and future are something observed relative to a particular dharma. Past is the transitive closure of all its causes, and future is the superset of dharmas directly or indirectly conditioned by a given dharma.

Ontologically, there is no becoming, no objects moving through time, nor one global time dimension.