Spreading of Chemicals (chemistry lab)
There is no sensible single answer to this: it depends on the chemical (and you really should read the safety notes before using anything or you will cause severe trouble.)

Consider these scenarios. 

You are working with a solution of common salt and you spill it on your lab coat. No problem at all; you could eat your dinner off the lab coat as it is the same slat you might sprinkle on food. But you spill a solution containing some mercury salts on your coat, get it cleaned and decontaminated immediately as it will eventually poison you even if you don't use it to wrap your dinner.

Or you spill some ether on your lab coat. Don't stand near naked flames until it has evaporated, but once it has gone, it has gone with no lingering effects. Spill some nitric acid/sulphuric acid mixtures on it and even if you wash it off before wearing it again you might still have the problem that it has created some nitrocellulose, which can spontaneously combust, taking you with it. Equally badly, you spill some tellurium salts on your coat. They might not kill you but you will be shunned for weeks as you will smell strongly of something like garlic, but more offensive.

Bottom line: read the safety instructions and act accordingly: there is no general rule.