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The separated liquid is whey. Whey is slightly acidic which gives it a mildly sour taste. Greek yogurt is just yogurt where more of the whey has been strained out.

To answer your question, there is certainly no harm in reincorporating it. I don't know if it will affect shelf-life at all, but I would expect if it did it would be by a negligible amount. Not reincorporating it may (over time) cause the yogurt to have a slightly less sour taste and a slightly thicker consistency, though you'd probably have to more actively strain it to get a noticeable affect. When I make yogurt, I just reincorporate all the whey which leads to a fairly runny but very tart yogurt.

Even if you don't reincorporate it, the whey itself can be used for flavoring in other foods, e.g. smoothies and sodas. It has some other cooking uses as well.