There have been many similar questions previously on this subject and so I have been experiementing. Last night I tasted a bottle of cheap red that had been first opened two weeks ago, then the screw cap replaced and kept in fridge since.
First. You won't be made ill. The wine is probably drinkable, tho' it may probably will not be as tasty as when you first opened it.
If you don't want to finish the entire bottle then recork it as quick as you can and put bottle upright in fridge door.
The cold slows down detrioration of the wine -- for the same reason why you keep other foods in the fridge.
How long the wine remains drinkable depends on the wine. basically the stronger and younger the wine is, the longer it keeps well.
Once a wine is opened oxygen gets to it and starts to affect it. Oxygen can be good in the short term -- that is why people decant red wines, let them 'breathe' and open bottles some time before drinking. It softens a tanninc wine. So it may be that a wine that you've kept in the fridge for three days actually tastes better than when first opened.
But at some stage the wine will deteriorate. This happens slowly, but gradually you'll find it is less fruity, gets a bit 'sharper' tasting, more acidic. It still won't make you sick. In fact it never will, you just will not want to drink it because it doesn't taste good long before the wine gets harmful - if it ever does.
If you expect to keep the opened wine longer than three days you could put it in the freezer instead of the fridge. Slowly thaw it before you want to drink it. It will taste about the same as as if it had been kept opened for three days.
Another option is to look for the single serving size bottles which are more common these days.
And what about the wine that had been in my fridge for two weeks? Well -- to be honest - I have had wine that tastes like that when I have bought wine by the glass in a bar. Thin and sour and I didn't want to drink any more of it.

But it won't make you sick.
Rule of thumb -- if it smells OK and tastes OK, it is OK.