OK, it's a topic that comes up not just for Georgia, but several countries. We were told by many sources that in Argentina paying tax was 'optional'. Lots of anecdotes about not really needing a work visa. That sort of thing.
However, when it comes down to it, the Georgian government certainly exists. And so do their police, and justice system. No doubting that.
And as we would have it, the Georgian Govt has a Visa web site. And on it, yes, in addition to tourist, spousal, visitor, residence and other visas, there is the Employment Visa.
So, legally and technically speaking, it is NOT 'permitted' on a tourist visa, because otherwise there'd be no need for the Employment Visa. The tourist visa permits access under which you cannot engage in business, use government benefits or study. And you cannot stay permanently. However, hypothetically you may get away with it. You might over-stay, and when you finally leave, encounter a lax border guard who doesn't care, and so you were 'permitted' to overstay. Similarly, you may be 'permitted' to work on a tourist visa, but if ever it comes down to it, an overzealous cop, an angry co-worker (or jealous local), it may not take much and you'd have nothing to defend against the written law, other than 'I was told it was ok'. Which unfortunately, does not work out as a defense.
That's a bit of a rant with some sense in there, I hope, but the point is - legally, no, but you may get away with it.