I guess it would night or shaded basically all the time for certain parts of the planet that fall just under the rings from the sun and what-not too.
I think those areas would have a day/night cycle similar to alaska, considering that the shadow from the rings would fall on a different place on the earth depending on what point in the earth's rotation it was.
So the northern hemisphere would have the band of shadow during its winter time, as would the southern hemisphere.
I'm not sure it'd impact plant life much, as plants go dormant during the winter time, and don't grow, so photosynthesis is much less important. I mean, grass lives even when it's completely covered with ice and therefore getting zero sunlight.
However I'd expect that the gravitational effects on the earth's tides would be constant. The equator would have more of the oceans due to the gravity being concentrated there, and so tides wouldn't be as high towards either end of the earth. I'm not sure that'd make life impossible, but I'm not a physicist or an ecologist so everything I'm saying here is conjecture based on what I've learned through school.