How long would it take for a full paint can to dry if the top is left off? Will it dry this way?
Depending on the type and amount of paint it might never truly dry. Here's why; as the top of the paint drys it will form a membrane. This membrane will thicken over time and it will also allow some of the solvent to evaporate. Eventually it becomes thick enough to drastically slow down and maybe even stop the solvent from evaporating.
Eventually the entire can will harden but it could take weeks or even months, especially if there is a lot of paint in the can.