How far do you push through injury? I'm not sure, but you should push. As long as you're not making things worse, I think you keep going. At least if you're a pushy person and driven. If you want to take care of your body, you have to keep pushing it, which often means dealing with pain. I've had a sore left elbow, maybe some low tendonitis on it for 2 weeks, a pulled and tight right hamstring for a couple weeks as well, and now a sprained wrist.
I've avoided lifting, but I'm still running, and I played baseball. Planning on playing this Sunday again. My thought is that I'm going to have a sore wrist for 6 weeks before it heals. That's if I do nothing with it. Likely that with some playing baseball, it will extend out to 10-12 weeks, but that's the price I'll pay since I don't want to stop and I don't think I'm making it worse. It's just healing slower.
I've had lots of sore muscles, sore joints (usually ankles) and more while on this running streak. I've dealt with the pain, gone slower, and pushed past it. And I've been surprised that over 2-3 weeks the pain will go away and things will heal. Pushing the body to keep going I think forces it to heal on its own.