Most organic solvents can be easily removed (boiling point, vapor pressure) by vacuum and a little heat.....water not!

As I see it water in small quantities forms minimum boiling azeotropes with quite a lot of solvents, and could even decrease bp of pure solvents and may increase vp.

Removing suspended water from organic solvents is necessary prior to the concentration of such solvents in rotary evaporators. Otherwise, the suspended water will keep with your compounds/extracts at the end of the concentration of organic solvents, and then removing it will be a very harder task!

In most of the cases its required to dry the organic phase. Further, as you said sometimes it may not be required where the carried water traces won't have adverse effect on your product isolation and quality. We need to decide on the basis of downstream process we operate. Still at research phase usually we dont take chance and thus we prefer drying.

In order to remove the water molecule that still exist in organic phase

please try to run an experiment extraction of a organic compound, for example with diethyl ether/H20 extraction. In one dry the organic phase, the other don't. See the yield and go for an IR spectra. Register and interpret the differences.

Water is having high boiling point than organic phase , if a little amount of water is still present with the organic phase , then there are also chances of leaching of extracted compound in water, then during evaporation of organic phase, due to high boiling point of water it will be evaporated in last and extracted compound will leach more inside water phase leaving the organic phase of decreasing volume . and also thermolabile compound might get destroyed in comparitively high boiling solvent like water. Even the characterization of product formed may be difficult in the cases where extract have not dried.

Many organic solvents are immiscible with aqueous solutions, but they are able to dissolve significant amounts of water because of their polarity. Unfortunately, water is a compound that is very difficult to remove from many compounds, because they are either holding on to it well (i.e., alcohols) or the compound itself is steam volatile. Bottom-line is that the more polar the solvent is, the more hygroscopic it will usually be because it dissolves the water better. Thus, removing water and other impurities from a solution can become an arduous task.

Depending on the next application of the synthesized compound, if proper characterizations (such as m.p, b.p. etc which are affected by impurities) and the next reaction is moisture sensitive, then complete removal of the water becomes important.

Besides using drying agents like sodium sulphate (dries slowly), magnesium sulphate (not used for acid sensitive compounds and dries faster) etc, the common practice is washing the organic layer with brine. Washing/shaking with brine pulls the water from the organic layer to the water layer as brine becomes more dilute. This will obviously reduces the residual amount in the organic layer that will be taken up by the drying agent.