Songs that start out sounding like they're playing through a tinny radio before bursting into full produced stereo

Portishead - Strangers.

Though it has a couple of bars of full stereo before going tinny-crackly and back again.

whatever the first song on blood sugar sex magic is called

Eels - Novacaine For the Soul

Weezer - Runaway

(if we can include dictaphone intros)

I'm trying to remember if the Monkees' "Magnolia Simms" ever shifts out of scratchy, old-timey mode.

No, it doesn't. There's even a note on the sleeve saying so.

Phil Ochs did a similar track around the same time--"Half A Century High" maybe?

not sure if Cut Copy - "Strangers in the Wind" qualifies

Slayer = Ghosts of War . also a tune on the new Soundgarden

Undertones - "Get Over You"

Ramones - "Do You Remember Rock and Roll Radio"

i don't know if "tiny" but Sonic Youth - Mote

sometimes i wish they made a full song out of that 2 seconds opening sound there

goofy's concern - butthole surfers

probably something by Man Or Astro Man?

One Big Family by Embrace.

First time I ever heard this effect was on the Chili Peppers' "The Power of Equality". I was 13 or so and that shit was like wow.

"If You All Get To Heaven" - Terence Trent D'Arby

aimee mann – 'amateur'
they might be giants – 'with the dark'

Kept trying to think of a Who song that would fit this and realized there isn't one. There's a whole album, though:
http://1.bp.blogspot.com/_M9Gmi7m9JCM/S7x8d6lfsLI/AAAAAAAAAFA/sa5uMKvgrhY/s200/concept09-The-Who-Sell-Out.jpg

"Don Caballero 3"

Ruins - Quetzalcoatl (first issue only? was checking out YT for audio clips and apparently the remaster doesn't include the muted drum intro)

Natalia Lafourcade & Gilberto Gil, "Farolito"

First time I ever heard this effect was on the Chili Peppers' "The Power of Equality". I was 13 or so and that shit was like wow.

the power of equalization

"The Love I'm Searching For" by the Rentals. I recently revisited this album and it's great! It's aged quite well.

Songs that start out sounding like they're playing through a tinny radio telephone before bursting into full produced stereo:

"You Burn Me Up I'm a Cigarette" - Robert Fripp

And of course "Telephone Line" - Electric Light Orchestra, but only the vocal

team dresch - screwing yer courage

Pale Saints - Way the World Is

Wings' "You Gave Me the Answer" has the "megaphone vocal," too.

"hummer" by smashing pumpkins, duh