To heat or not to heat; distillation?
You obviously already have a general idea of what is going on in a distillation, so I will spare you the picture and everything.

Your question can be broken into two parts:


Do I need heat for vapourisation?
Do I need cooling for condensation?


I am going to address them one after the other.

Vapourisation

As you probably also know, reducing pressure is one means of evaporating liquids. This is used in rotavaps which use a $40~\mathrm{^\circ C}$ bath and reduce the pressure to make sure the solvent evaporates. You could think that just reducing the pressure should be enough to vapourise any liquid.

But if you vapourise a liquid only by reducing pressure, you will reduce the temperature of the liquid. This is the evaporation cooling that your body uses when sweating: If the water in sweat evaporates it cools the surface it evaporates from. Thus, after a certain time and especially if you attempt to distill large quantities of water, it will freeze and further evaporation will become exponentially harder. Therefore, a practical setup will always require heating.

Condensation

Effectively, this is the same discussion but in reverse. Note that if you boil water in a closed pot, the lid of the pot will get successively hotter due to condensing water releasing heat. After a certain time, the temperature of the surface you want to condense on will be so hot that the liquid will no longer condense.

This is especially a problem if you decided to reduce pressure for evaporation: If you have to reduce the pressure so far that the liquid would evaporate at $10~\mathrm{^\circ C}$, then there is no way that it will condense at a cooling trap of $25~\mathrm{^\circ C}$. Therefore, the condensation area must be cooled which is typically achieved with Liebig coolers or similar.

What about vapourisation followed by condensation in the same vessel with changing pressure?

Well, that is possible. It is not practical, though, since the inside of a single vessel can only have a certain pressure (within slight margins). And vapour spreads out uniformly. Your vessel would have to have an initial ‘bowl’ and a collecting one. You reduce pressure to evaporate out of the initial bowl and then re-raise pressure. The liquid will condense uniformly and you will lose a significant amount of it because it will drip back into the initial bowl. Your yield with a typical common setup is much better.