What cognitive bias prevents you from discarding stuff? Let's say person is moving homes and has lots of kids toys, suits, shirts, ties and other stuff that he seldom uses. 

He is am planning to sell it all in a car boot sale even if he knows that it's counter productive. He would have to use ~30£ on entry fees and fuel, without even accounting for Sunday lost. 

Logical solution would be just drive to the nearest skip or charity shop, and yet he can't make himself to do it, which of cognitive bias is at play here?

I don't think this is 'sunk cost' because he is loosing that stuff anyway, it's just that loosing it and knowing that someone else will be able to use it somehow makes ones loss more acceptable, which is sub-optimal and suggests a bias.
One possible explanation for the scenario you are describing is Endowment effect. 


  In psychology and behavioral economics, the endowment effect is the
  hypothesis that people ascribe more value to things merely because
  they own them.


One of the theories that explain the phenomenon is based on loss
aversion : 


  It was proposed by Kahneman and his colleagues that the
  endowment effect is, in part, due to the fact that once a person owns
  an item, forgoing it feels like a loss, and humans are loss-averse.