I pretty much have a chainring tattoo from all the times I've "gotten" myself, but never had it prevent me from riding. Prevent being "too painful" and not in a "be a man" sort of way. Mostly superficial cuts in my case.

Chainring cuts are typically puncture wounds, if you've not had a Tetanus shot in a while, you might want to see a doctor. Also, if the wounds are red and inflamed, you might want to seek medical care.

The soreness could also be from a strained/sprained muscle and not be directly related to the cuts.

Moral of my story: better safe than sorry, check with your doctor.

check with the DOC, esp if you don't have active Tetanus!!!
Ice, Compression, Elevation -->for any traumatic injury. And as a Rule of Thumb: first 100hours --> ice it.
the actual abbreviation is RICE (rest, ice, compression, elevation),but I usually ignore "rest" if the pain is sustanable and not very important structure is injured. It's totaly up to you.
Check with the MD,
Good luck,
Olga