Carrion birds, such as crows and vultures, eat dead animals. As a whole, it's possible that they detect carrion by the smell or visually.
Crows specifically are omnivorous and predatory in addition to being carrion scavengers. I suspect that they will try to eat anything that doesn't fight back, especially if they are hungry enough and there is exposed flesh. I don't know if crows prefer dead meat or fresh meat, but they certainly eat both.
Suppose you tied up someone and left him on the field. Would that attract the crows?
The crows would probably be attracted because they are curious and want to check out the new, unfamiliar object. They would certainly be able to tell that you are alive by body temperature and sounds of breathing or heartbeats.
I think they will be hesitant to eat you because live humans are unfamiliar to them. They will probably find other carrion nearby and eat that instead. If you stayed unconscious for several days, the crows would get used to you, lose their fear, and eventually one would go for a taste. Once the first gets a bite or two, and you fail to retaliate, I think the others will quickly join in.
Suppose this person fell down, was bleeding, and immobile, but not seriously injured? Or is it the death part?
The fact that they can see and smell the meat, and don't have to bother piercing the skin, would make you more attractive to the crows (supposedly, they like to go for things like eyeballs even on live prey).
However, even if immobile, a conscious human could struggle (shake their head, yell, roll around) which would startle the crows and keep them at bay for a time. If you can't actually interfere with them pecking you, I'm sure they would eventually get over their fear and attack you. If you can physically swat or shake off the crows, they'd just wait for you to get tired or die, which probably won't take too long given the injuries.
In conclusion, I don't think it's impossible for crows to eat bits of a live human, whether conscious or not. However, for various reasons, it would probably take a few days for them to actually get around to it. If you are not dead by this point, larger animals would show up and attack you, and crows would end up picking at the scraps.