Nothing at all will happen. Why do you think "Andy" is shaking his head at all of the BS you guys are feeding yourselves. .003 just isn't enough to worry about. As he stated, the only time it may give you any trouble at all is if you have a match chamber and that's a maybe.
Shoot them. They will either shoot well or not. Your ball won't fall off, hair won't grow on your palms and it won't effect your vision. You won't get pregnant.
Those bullets may just be the best buy you've made in a while or they may not. If your rifle is in such questionable condition, which I doubt, there are other issues to address than a projectile that is grossly underweight and soft cored to boot but a few thou oversize. Load them hot if you want to, it will be next to impossible to get pressures high enough to create any worries. Have fun. It's not rocket science or voo doo.
Think about it this way, the Spanish loaded them into weaker actioned 93 mausers at velocities that were acceptable to NATO and with terminal results within specs. Why would you worry about it in a modern rifle?