Is this exposed wire dangerous?
Given the colours (I don't know why the other answerer referred to a red wire, the wire in your picture is clearly brown) I'm going to assume you are somewhere in Europe.

The insulation on the nominally live wire is gone, that means you more likely than not have 230V exposed (if you are in a country that cares about polairty you definately have 230V exposed, if you are in a country that doesn't care about polarity it's 50/50). Touching that is very likely to result in an electric shock, if you are unlucky said shock may be fatal. 

There is also earth exposed. With some movement you could end up with a short circuit. Most likely this will blow a fuse/trip a breaker but if you are unlucky it could also start a fire.

I would advise against using insulation tape to repair mains wires except in a dire emergency. It insulates fine but the adhesive can lose stickiness over time. If you do insist on using it make sure you remove the outer sheath, insulate the damaged wires individually and then insulate the bundle.

Best option is to replace the cable, if that is not possible then it may be possible to cut it and insert a suitable joint.