And we all know what “Lucy in the Sky with Diamonds” is all about…
Lord of the Rings!
Picture yourself in a boat on a river
Undoubtedly a reference to the river Anduin, down which the Fellowship sailed after leaving Lothlòrien.
With tangerine trees
Mallorn trees, which had gold leaves in winter, at the time the Company of the Ring left.
and marmalade skies.
Refers to the red sky over Mordor, which became increasingly visible as they approached.
Somebody calls you,
Nazgûl calling to Frodo as he wore the Ring: “Come back! Come back! To Mordor we will take you!”
you answer quite slowly,
As Frodo struggles with the temptation to put on the Ring.
A girl with kaleidoscope eyes.
Frodo saw the Eye of Sauron in Galadriel’s Mirror.
Cellophane flowers of yellow and green
Refers to the yellow flower elanor, which grew in Lothlòrien.
Towering
Many towers figure prominently in LotR – based on the context, this may be a reference to the Argonath, or Pillars of the Kings.
over your head.
Because hobbits are short, and the Argonath are very tall anyway.
Look for the girl with the sun in her eyes
Galadriel again: “the light of the Two Trees shone in her eyes.”
And she’s gone.
Either because they’ve left Lorien, or could be a reference to the disappearance of the “real world” when Frodo wore the Ring, or to the departure of the Elves from Middle-earth. Or all three.
Lucy in the sky
A confusing reference – may be related to Varda, or possibly to Eärendil, though the feminine name Lucy makes this unlikely.
with diamonds
However, this obvious reference to the Silmarils increases the parallel to Eärendil.
**Lucy in the sky with diamonds
Lucy in the sky with diamonds, ah, ah
Follow her down to a bridge**
That is, the bridge of Khazad-dûm.
by the fountain
The Mirrormere. Also recalls the Mirror of Galadriel.
where rocking horse people
Obviously the Rohirrim.
eat marshmallow pies.
Perhaps lembas – an odd juxtaposition recalling the aid the Elves of Lorien gave to Eorl the Young.
Everyone smiles as you drift past the flowers
That grow so incredibly high.
Recalls the dreamlike atmosphere of Lorien (with perhaps another nod to the stature of Hobbits).
Newspaper taxis
Grey elven-ships, the color of newspaper.
appear on the shore
The Grey Havens, on the Western shores of Middle-earth.
Waiting to take you away
To the West, that is.
Climb in the back with your head in the clouds
Cf. Frodo’s departure from Middle-earth, where Tol Eressëa was revealed to him (and the others on the boat), but those on the shore only saw clouds and rain.
And you’re gone.
This line speaks for itself – those who went to the West would never return.
**Ref.
Picture yourself on a train in a station
With plasticine porters with looking glass ties,**
An image of mundane life, undoubtedly that of the reader.
Suddenly someone is there at the turnstile,
The girl with kaleidoscope eyes.
But once we pick up the book, any of us can travel to Middle-earth.
Ref…
Yes, I did make all of that up, of course. 
“Stairway to Heaven” and “Battle of Evermore” really do have LotR echoes – according to Robert Plant, the ones in “Stairway” aren’t intentional.