Like much with technology (and life), it depends. You didn't specify what software, protocol, or codec you are streaming with, so there are quite a few variables.
Will the bit rate drop?
If you are using a constant bit rate (CBR) codec, no it will not. As the name implies, the bit rate is always the same regardless of the content.
If you are using a variable bit rate (VBR) codec, it might be lower than when the lights are on. It may also not be significantly lower, because, as the other answer alluded to, you will get a lot of noise. When the light level drops, the camera will automatically increase the gain which will result in color artifacts in the dark area, like in the sample video below. It is very difficult for codecs to handle noise, so you might actually see an increase in the bit rate over a brightly lit scene with little movement.
Will the latency drop?
If you are using a codec that has several latency modes, then the it might. However, most of those codecs are commercial products that are either a proprietary codec or fudge things in incompatible ways within the standard. That makes it unlikely that you are using one of those codecs. On top of that, those codecs usually make you select the latency mode before starting the stream and it is then fixed while the stream is up.