Thanks to Alan Marchand for submitting the initial text and list below.
If you know of other exceptions, please send them to me!

This list is based upon a "rule" that I learned as a student of junior high school English:

"I before E except after C; exceptions are either, neither,
foreign, leisure, protein, seize, and weird."

This list of rule exceptions strikes me as being woefully inadequate. To get the ball rolling, I am offering some additional entries, as follows. Some of the foregoing have their origins in other languages [e. g., treillage (French), leitmotiv (German), etc.]. Nevertheless, these words appear in English dictionaries, so they qualify for inclusion in the list of exceptions below. If names are included we can add Keith, Sheila, and Neil. Note that the opposite also applies: the words "sufficient" and "ancient" have "ie" after a "c."

Acleisanthes (a genus of desert plants)
ageing
agreeing
ancient
apartheid
atheist
beige
being
blueish
caffeine (and also decaffeinated)
canoeing
casein
ceiling
cheiloplasty (lip reduction)
cheiloscopy (lip prints)
cheilosis (disease caused by riboflavin deficiency)
cleidomastoid (collar bone)
cleistogamy (the ability of some plants to propagate via self-pollination)
codeine
conscience
counterfeit
cysteine (a semi-essential proteinogenic amino acid)
deicing
deify, deifying
deign
deism, deist, deity
deionized
dreidel
eider (as in waterfowl)
eidetic (photographic memory)
eigenvalue (and also eigenmode)
eight (and related numbers, such as 1/8, 1/80, 18, 80...)
either
epeiric
epeirid
eyeing
feign
feint
feisty
financier
fleeing
foreign
forfeit
freight
geisha
guaranteeing
heifer
heir, heirloom
height, heighten
heist, heisted
herein
inveigh
kaleidoscope
keister
kneeing
leiomyoma (uterine fibroid)
leisure, leisurely
leitmotiv
madeira
meiosis
mieskeit
monotheism
neigh
neighbor, neighborhood
nonpareil
obeisance
ogreish
overseeing
plebeian
Pleiades (star cluster)
pleiotropy
Pleistocene
poltergeist
prescient
protein
reign
rein
reindeer
reinforce, reinstall, preinstall, reincarnation, reinvent, reinstate, reinstitute
  (plus any other "re-in" words such as "reignited" with the "e" and "i" in adjacent syllables)
reiterate
reveille
science
seeing
seigniorage  (or seigneurage, if you prefer)
seigniorial
seine (as in "net")
seismic
seitan
seize, seized, seizure
sensei
sheik
skein
sleigh
sleight (of hand)
sovereign
species
spontaneity
stein
surfeit
teleidoscope
their
theist
therein
treif
treillage (as in "trellis")
weigh, weight
weir
weird
wherein
veil, veiled, unveiled
vein
zeitgeist (bonus double exception!)

More from Alan:

My English teacher taught me well
A rule designed to help me spell
All words that range from A to Z
With 'I' 'fore 'E 'cept after 'C.'
This simple phrase, you'll be so kind,
To ever after bear in mind
And thus ensure the right decision
When writing words with great precision.

Santa's reindeer deign to neigh
When faced with overloaded sleigh.
An ancient species, gone astray,
Shall ever on my conscience weigh.
I could continue in this vein
As long as mass confusion reigns.
Another rule of thumb, I'd say,
That feisty neighbors oft inveigh.

This poem by Jef Raskin includes several dozen exceptions:

"I before E
Except after C,
Unless pronounced A
As in 'neighbor' or 'weigh'"

Education is forfeit for reinforcing such rules!
Sound a feisty reveille while eyeing the schools!
Neither will our heirs be agreeing to deceptions
Once seeing, herein, these sufficient exceptions:

We were seized by a feeling
For fleeing on the ceiling
To a leisurely meal
With Keith, Sheila, and Neil

We drank madeira, so foreign, in steins
Along with a surfeit of weird blueish wines
Being foolish, took codeine, ate ancient proteins
Therein guaranteeing these ogreish scenes

Wherein we're canoeing to a new sovereign state
While deicing a kaleidoscope on a hot jadeite plate
And kneeing obeisance to an overseeing king
Our plebeian lips kissed his counterfeit ring.

Then we unveiled their sleight-of-hand trick
Deifying a heifer, with effect atheistic
And falling from the heights with a loud seismic crunch
We reignited the nonpareils we had heisted for lunch.

So I before E
Except after C
Unless pronounced A?
False decreeing, I say!

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