Can you long rest during Astral Projection?
The implications of this are that you can simply astral project, travel through the Ethereal Plane back to the Material Plane through the Silver Color Pool as if it were a foreign plane, and then live out your life as an immortal with your body in a closet somewhere safe.
Without the ability to long rest, eventually your ability resources would run out, so you'd have to drop Astral Projection, get some rest, and then recast the spell with its costly components.
Similarly, if you can't long rest during Astral Projection, can you short rest?
The above would now apply, but to Warlocks and Monks exclusively as their ability resources are refunded on short rests. The difference being they would never regain hit die and there are some daily resources for Warlocks which aren't reclaimed.
According to the spell description, that wouldn't work.
The moment you return to the Material Plane, your body snaps to you- you can't reside eternally on the Material Plane in your astral form.
While you're in your astral form, you don't age or feel the effects of hunger, but no other effects are given, save one:
You could thus stash your body in a closet, astrally project, and become immortal in the Astral Plane or Outer Planes- or you could teleport about the Material Plane, Ethereal Plane and Inner Planes by means of astral projection, but only in the Astral and Outer Planes do you have the protection inherent to astral projection- most likely because those are the planes where you do not exist in a wholly physical form.
As for resting during astral travel:
It should be noted that several features of the Outer Planes take effect when you complete a long rest (notably psychic dissonance), and nothing about the Astral Planes says that you cannot take a rest, so you can indeed take long or short rests in astral form. In order to be kept from taking those rests, the spell description would need to specifically prevent you from doing so.