Q:

Are photons eternal or do they really cease to exist?

I have read this question:
Are all electrons identical?
Same photon or different photon?
What happens when a photon "dies"?
But these do not give satisfactory answers.
When a photon is absorbed, we usually say that the photon ceases to exist as photon, and transforms into the energy of the absorbing electron/atom system.
There is a very good description in this answer from John Rennie:
Same photon or different photon?

Your question is based on the assumption that a photon is a fundamental object i.e. that photons are something we can point to and say here is photon 1, here is photon 2, and so on. The trouble is that quantum field theory particles are somewhat elusive objects. This is particularly so for particles like photons that are their own antiparticles because such particles can be freely created and destroyed. At least fermions like electrons are protected by conservation of lepton number.
  In general energy propagating in a quantum field looks like a particle only when energy is being transferred into or out of the field i.e. when a photon is created or destroyed. Outside of these events it's hard to point to anything that looks like a photon.

So basically photons are QM objects, excitation of a field, and in certain cases we say that this QM object ceases to exist. 
So basically there are two cases, partial energy transfer (inelastic scattering), and full transfer (absorption).
When a photon is inelastically scattered, it transfers part of its energy to the other field (maybe the electron field), but we do not say that part of the photon ceased to exist. But then this partial energy that was transformed, can again be emitted as a new photon. This is part of the reasons why photon number does not have to be conserved. Only energy is conserved.

So the photons are absorbed and disappear when hitting the leaves of the tree. It is energy that is absorbed and conserved and released as photons when burning.

Are the photons released by trees the same generated by the sun?
Energy is conserved, and eternal. It gets transformed, propagates. Photons carry energy and are the very quanta of light. If energy is eternal, then we could say that photons are too existing eternally, sometimes they get absorbed fully or partially, stored in other fields (like the electron field), then re-emitted.
Question:

Are photons eternal or do they really cease to exist?

A:

No, in modern physics we don't think so. Quantum field theory allows the creation and annihilation of particles (e.g. in pair production) so the number of a given particle, e.g. the number of photons, doesn't have to be conserved. Energy does have to be conserved, but it can be transferred to other particles, so photons can be absorbed. We can say that the photon ceases to exist because the electromagnetic field ceases to be excited.

