No, you don't have to let it dry. In fact you shouldn't.
Thermal paste provides better contact between the chip and the heatsink. The heatsink, and to a lesser extent the CPU core, isn't completely flat. They are textured, and that makes it impossible to get total contact between the CPU and heatsink, even if these spaces are microscopic. Air doesn't conduct heat very well, so thermal paste fills up these cavities and provides better contact and conduction. I strongly advice against even powering up a computer that hasn't got thermal paste (or that thermal tape) applied between the CPU and heatsink.
Remember, a very thin layer is enough! Too much and it will actually insulate instead of provide better conduction.