What is the effect of substituting oxygen with sulfur in drug molecules?
In asking this question - Why do chalcogens (Group VI) stink so badly? - I received a couple answers and comments along the lines of "it's not quite so simple". The same would generally apply here; there isn't one change in behavior, or even a finite set of behavioral changes, that could be predicted by an oxygen-to-sulfur exchange.

Just as a basic example, compare methanol ($\ce{CH_3(OH)}$) to methyl mercaptan, ($\ce{CH_3(SH)}$). Methanol is a colorless liquid at RT with a sweetish odor. Methyl mercaptan, with a higher molecular mass due to the sulfur switch, is nevertheless a gas at RT and smells like rotting cabbage. Methanol metabolizes into formic acid and formaldehyde which in sufficient doses causes nerve damage, specifically to the optic nerve. Methyl mercaptan is pretty much its own metabolite; while also toxic in sufficient doses, it's readily filtered out by the liver and kidneys, as it's produced by bacterial breakdown of many of the foods we eat (especially highly-fibrous vegetables, legumes etc).

Perhaps the most common occurrence of sulfur in otherwise organic molecules is a sulfhydryl group (R-SH), forming a thiol. The alcohol-to-thiol switch is relatively common in human biochemistry and as such is also of interest in pharmacology. The Wikipedia page on thiols lists the following basic changes between an alcohol and its thiol analog:


More easily oxidized, producing disulfides, esters, ethers, etc
As thiols, more acidic than alcohols
Contrasting this, thiols dehydrogenate easily to form thiolate bases
Thiolates complex strongly to most metals


In general, you can expect the sulfur analog of pretty much any organic functional group (alcohols->thiols, aldehydes->thials, ketones->thioketones, esters->thioesters, ethers->thioethers, etc) to be more reactive. However, exactly how the sulfur switch will affect the actual reactions the compound will undergo is best studied case by case, as there is no one way that a particular sulfur-bearing functional group will behave when introduced into the complex chemical stew of human biology. 

One near-constant; sulfur in small doses can smell pleasant, because sulfur is an essential not-so-trace element in humans (oranges and grapefruit, for instance, have some volatile sulfur compounds that contribute to their smell and taste). Sulfur in high concentrations stinks, because for the same reasons it is essential in biochemistry, too much of it is toxic (which is a pretty good illustration of the First Rule of Toxicology - every substance, in sufficient dosage, is toxic).