You say that heat still touches the earth, but light is blocked. This is interesting because it might not cause what you think to happen.
So, the main thing to look at is the black-body curve for radiation from a body at ~6000 degrees Kelvin (AKA Our Sun). http://www.ces.fau.edu/ces/nasa/images/Energy/RadiationIntensityVSWavelength-690x509.jpg
If we reinterpret your question to say that all visible light is going "under the shadow", then most of our radiation is in that band and we've got a reduction in the 44% of energy from the Sun coming our way, but the remainder is unaffected. That's a pretty big fall and you can understand what happens when radiation is blocked out by previous mass extinction events due to volcanic eruptions and asteroid impacts (e.g. the impact that wiped out the dinosaurs).
If we get rid of 100% of visible light (not what you're asking and worst case, but useful to think about), we get a very severe nuclear winter, with less severe effects the more light you let in. Twilight probably implies about 30-50%, just guessing from angles, so that would still be quite severe.
But that's not all that happens. There's also albedo to consider. The Earth reflects about 0.3 of what gets thrown at it. If the earth is covered in shadow, this can cause nuclear winter, which causes the world to get covered in snow and the albedo to go up and reflect even more light, causing temperatures to drop even further in the Snowball Effect.
On the other hand, shadow is black and black is a very effective colour for absorbing radiation (albedo=0). You could actually end up with things getting warmer in the atmosphere by the presence of shadow as you nearly double the absorption of heat of the Earth (assuming your shadow is not just reflecting it away).
That's not all, because we need to think about how and where things are reflected and absorbed. The atmosphere is good at absorbing some wavelengths of light, but not others, and the ground will absorb most light and reflect it at lower wavelengths. See this link: http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap7-14.gif for an idea of what the atmosphere absorbs on the way down. Mainly, it absorbs a lot of heat radiation (infrared).
This leads me on to the Greenhouse Effect (mentioned in the article I took the picture from), where the atmosphere absorbs most of the emitted radiation from the ground, making us toasty and warm. There's a simple calculation they do in that article which talks about what temperature the Earth would be if there was just albedo and normal radiation involved, and it's a cold 255 degrees Kelvin. http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap7.html
Mostly, visible light adds to the Greenhouse Effect and that is precisely what you're cutting out, which would lead to a huge drop in temperatures (assuming your shadow isn't absorbing light anyway).
All of this leads to some very interesting weather effects if the atmosphere gets a lot warmer than the ground (which I am not qualified to talk about). Usually, we have weather patterns dominated by air close to the ground getting warmer, rising and then cooling, dropping down in other places (these places usually are our deserts). However, cutting out visible light would affect that greatly.