If I don't use a torque wrench, how do I tighten to the approximate torque? What can go wrong?
You can't approximate torque by feel. Loctite, grease, carbon compounds can all alter the 'feel' when you are tightening down bolts and if there was anyway to 'approximate the feel' these variables render it fairly impossible. 

If you use ti/aluminum bolts, carbon components, high end aluminum do not cheap out, get the torque wrench. 

If you tend towards stainless/steel bolts and less expensive components, you can forgo the torque wrench and just ensure you are working to even torque. For example, if it is a faceplate make sure that the gap between faceplate and stem is even all the way around.  

Saying 'too tight is better than not enough' is not true. Too tight can create stress risers in a component (specifically I am thinking handlebar) that can lead to catastrophic failure down the road. Too tight on a square taper crankset (specifically the fixing bolt) can cause deformation of the taper fit and eventual loosening of the crank. 

Hopefully that helps. 

Noticed I didn't answer the question fully.

Faceplate/Bar interface
Stem/Steer interface
Crankset (whether pinchbolt or square taper style BB)

Those are areas where I would ensure proper torque. Seatpost is dependent on style of bike (ie. Madone seatMAST, or carbon post/carbon frame) as to whether torque is super important. Rotor bolts, brake mounting bolts etc... but with increasing importance as exotic materials increase.