This really depends on your protocol.
But usually DNA extraction is done at room temperature (if using a commercial kit).
The only step that is kept on ice is pellet resuspension step. It contians an enzyme called RNase. RNase degrades any RNA after the cells are lysed.
To deactivate endonucleases, high salt concentrations should be fine. I guess (I can only guess since I do not know your protocol) the ice temperature is to deavtivate any enzymes and/or aid in precipitation of DNA, once the cell debris is taken out.