Mine did/does. I wore my H's white tshirts to bed and stained them. Everyone said breastmilk doesn't stain. I have sooo many ruined baby clothes and A&A blankets.
It depends on your diet. Things that stain make your milk stain (like beetroot and that yellow artificial dye), things like sushi will tint your milk but not stain IME.
Everything stained for me- milk, lanolin, spit up. I didn't stress about it and will just junk all my nursing tanks when I'm done. I never used to wear a bra at night, but now I wear a sleep bra and probably will never go without one again because my boobs just aren't the same. Plus, in the beginning, a bra or tank will keep your shirt from rubbing against your nipples. I got the nursing sleep bras from Target and I love them... the material is so silky, it feels really good when your nips are sore!
I have a tip on this subject though. We got a bunch of crib mattress pads and they're smaller/easier to wash than our queen size mattress pad, so we lay those down in the appropriate boob region under my sheet but on top of my normal mattress pad and wash that every time we change sheets. At first my husband kept laying down burp cloths, but we moved around too much in bed for them to stay put and one day I woke up on an excessive pile of eight prefolds.
I do something similar with the waterproof burp cloths. I'll lay a large one under me and baby. I feed her in my bed at night and we are messy.