In my limited testing I have found:
1) Shrines seem to only be able to be used 3 times per run.
2) "Tutorial Smith", the dying robot you encounter at the beginning of the game will give you hints about shrines.
3) When you get a random equipement change it doesn't show the new equipment on your stats screen (select button on gamepad)

Achievements:
Lil' Tyr
Donate 27 gray scrap to shrines.

Little Zurvan
Donate 125 gray scrap to shrines.

Little Phaestus
Donate 216 gray scrap to shrines.

Lil' Wadjet
Donate 343 gray scrap to shrines

Lil' Buluc
Donate 512 gray scrap to shrines

Lil' Orphy
Donate 729 gray scrap to shrines

Holy number: 3

Unlock:
An orb bot follows you, eating scrap. When it consumes enough the player becomes invincible for a few minutes. If the player comes into contact with an enemy it takes damage.

Overview:
Most of the effects applied due to offers are temporary.

Offers (good)
33 grey, 1 red: aura bolts, your aura now has a damaging aura.
57 grey, 1 blue: big bolt, your bolt is now quite large.

Offers (ok)
21 scrap: Your offer pleases me. A temporary boon to the might of your weaponry.
24 scrap: Your offer pleases me. A temporary boon to the power of your weaponry.
69 scrap: Your offer pleases me. A temporary boon to the might of your weapon.
This was a +6 boost to my weapon power that lasted roughly a minute...
99 scrap: your offer pleases me, a temporary boon to the power of your weapon.
3 green: a temporary boon to haste

Offers (bad)
0 scrap: You offer me nothing! A temporary curse on the power of your weaponry.
1 green: Your offer offends me, a temporary curse upon your rate of fire.
2 blue, 1 green: this is what you offer me? There is nothing I can do for you.

Holy Number: 5

Offers (good)
5 grey, 1 red: I grant longevity to you. +1 to health
5 grey, 1 blue: I grant longevity to you. +1 to health

Offers (ok)
15 grey: heals you.

Holy number: 6

Offers (good)
12 grey, 1 red: energy vorb, converts meat to energy
12 grey, 1 green: you get chrome dome, allies inspired to do more with less.
12 grey, 1 blue: Lil' Phaestus
18 grey, 1 red: you got power pauldrons, energy regen increased.
36 grey, 1 red: you got scrap orb, converts meat to scrap
36 grey, 1 green: you get chrome dome, allies inspired to do more with less.
36 grey, 1 blue: you get chrome dome, allies inspired to do more with less.
1 red, 1 green, 1 blue: Your offer pleases me. Thrice I enhance your holy power"
increased my energy by 3.
1 red, 2 green, 2 blue: A temporary boon to your strength.

Offers (ok)
66 grey: refills your energy

Offers (bad)
nothing: how dare you offer me nothing, i strip you of your health!
1 scrap: "your offer angers me, I strip you of health, fight smith 5."
15 scrap: "your offer angers me, I curse you with weakness in all ways, fight smith 5."
99 scrap: "your offer offends me, i strip you of your power"
appears to have set my energy to 0.
25 grey, 1 red: your offer offends me, I make you weak in all ways
1 green: "ill give you what you need"
3 red: "this is what you offer me? I will remake you"
3 blue: "your offer offends me, a temporary curse upon your rate of fire"
3 red, 3 green, 3 blue: "this is what you offer me? I will remake you"

Holy number: 7

Offers:
Haven't tried any yet.

Holy number: 8


Offers:
10 scrap: got nothing
13 scrap: gave me an "array of scrap" but my inventory had nothing new in it...
14 scrap: made me better
14 scrap: randomized my stuff
16 scrap, 1 red: tri orb
16 scrap, 1 green: machine gun
16 scrap, 1 blue: scatter gun
30 scrap: your offer pleases me, a temporary incrase to the speed of your bolt.
49 scrap: This is what you offer me. A few alterations to your equipment.
50 scrap: This is what you offer me. A few alterations to your equipment.
50 scrap, 1 green scrap: your offer offends me, I strip you of your energy.
88 scrap: attack rate up +1.
99 scrap: Your offer pleases me, I invigorate your fighting hand with speed.
1 green scrap: this is what you offer me? I shall remake you.
1 green, 1 blue: Nothing I can do for you.
1 red, 1 green, 1 blue: Remakes you.

Holy number: 9

Lil Orphy unlocked at 729 scrapped offer.

Offers (good):
9 scrap: "your offer pleases me, my children protect you"
18 grey: 9(?) nanobots added
81 grey: 3 temporary large purple helmeted orbs
99 grey: 3 temporary large purple helmeted orbs
9 grey, 1 blue: Shield orb
18 grey, 1 red: mothorb, converts scrap to nanobots
27 grey, 1 red: auto turrent orb
27 grey, 1 green: health orb
27 grey, 1 blue: buzz saw orb
3 red, 3 green, 3 blue: +1 to all stats
54 grey, 1 green, 1 red, 1 blue: Health orb
54 grey, 1 green: Explorb

Offers (bad):
2 scrap: "your offer offends me, thrice I strip bolts from your violent hand"
5 scrap: "this is what you offer me, I remake you"
9 grey, 1 green, 1 red: "Your offer pleases me, twofold error"
1 red, 1 green, 1 blue: "I can do nothing for you"
6 grey, 1 red, 1 green, 1 blue: "I can do nothing for you, an array of scrap"
2 red, 2 green, 2 blue: "I can do nothing for you"
3 red: "this is what you offer me, A few alterations to you"
42 grey, 2 green: your offer please me, a temporary boost to the speed of your bolt"