Is it healthy for no frost fridge to have water in ice cube forms? It's said that we should close liquids when we put them into fridge that has no frost system (no ice formation in the freezer).Or ice will appear and the fridge will not serve that long. 
At the same time there are forms for ice cubes in my fridge, in the upper shelf of the freezer.
They're open, so I supposed to pour water and leave this water exposed to the freezer.
I think if I'll use them I'll harm my fridge, I'll have to defrost it or may be even fix it.
In short, it's within design tolerance.
Frost-free freezers work by generating a periodic warm-up phase which lets any ice melt. The water is then drained to near the heat exchanger, which will then evaporate it off outside the freezer body.
Ice cube trays don't generate sufficient additional water vapour within the freezer compartment to compromise this process - except in the runners directly surrounding the trays themselves, which for some bizarre reason don't seem to ever stay as frost-free as the rest of the compartment;)