I am thinking about doing some streaming on Twitch, and I thought I would use some equipment I already have instead of buying new. I have a Sony alpha 5000 DSLR camera. Would continuous video capture (10 hours or more) damage the camera in some way? I am thinking of having it on a tripod, and filming my 3DS.
It depends on the camera. Digital sensors such as those used in DSLRs produce heat as they capture images. The main issue with extended video capture is the accumulation of heat in the sensor. If it gets hot enough, it is possible that either a) quality can be reduced temporarily due to increased noise or b) at the extreme end, damage to the sensor could result.
In many dedicated video cameras, even high end professional models, fans or other cooling solutions are used to keep the sensor cool, but smaller sensors (such as in devices like gopros) are able to cool from simple convection.
Most modern cameras have temperature monitoring and can shut down when they get too hot, so you are unlikely to cause damage, but you might not be able to film as long as you'd like to. The environment you are shooting in will also make a big difference. If you are shooting inside in a cool room, it will be much easier for the sensor to keep itself cool than if you are shooting outside on a hot day.
Other than thermal issues, I can't think of anything other than just the general wear on the sensor. Video obviously requires far more constant sampling of the sensor, so it has the potential to wear the sensor out more quickly, but the sensor was also designed for video and it's unlikely that the sensor will die due to use prior to the camera being significantly outdated, so it probably isn't a major concern unless you are planning on running it 10 hours a day, every day, for a long time.
With the Sony alpha 5000 camera specifically, you will run into trouble trying to do continuous video capture (10 hours or more).
It will stop recording after 29min and 50seconds and then you will have to start it recording again.
Update: Per @John this only applies to actual recording. If you plugin the HDMI out to a video capture device, you can stream the live viewer-display for as long as the battery lasts.