Clean up means to tidy or clean completely (I agree with Peachey). It could also imply that something was very dirty or messy before cleaning, whereas you could "clean" something even if it was not very dirty before.
Eat up means to finish eating for example I am going to eat up the cake means I started eating the cake last week/yesterday and now I am going to finish it. Or a mother might tell her child to eat up their (i.e. finish) the food on their plate.
Open up means opening up a box if it is difficult to open or intricate in some way ("open" does not imply any difficulty or special kind of opening), or a person can open up emotionally if they tell you about their feelings.