What is the plain tube (next to the breather with the yellow tip) that one end empties out near the bottom of the frame and the other end leads to what looks like the float bowl? For some reason it has started dripping fuel overnight. I have had this bike for six months (always left the fuel tap on) it has never leaked fuel. It was on a level stand (as I have left it many times after lubing the chain). I moved the bike outside turned the tap off and it seems to have stopped dripping and all dried out during the day but leaves me a bit bemused.
It is not the lower breather pipe (with the yellow one-way valve) that seemed to be dripping I think it was more likely the plain tube which from your description must be the "overflow tube that comes off the bottom of the carb and feeds down between the frame and swing arm".
So if it is just an overflow tube, any ideas why it has started dripping fuel out of the blue?
I guess the answer is that the float needle wasn't seating and sealing properly. Could be that a little bit of dirt got on the needle or else it is worn or damaged.
You could open the drain valve at the bottom of the float bowl and flush a bit of petrol through to see if that clears things.
Else you may need a new needle valve set - part number 4GY-14107-00-00.
It might also be possible that you have the float level set too high and the leak occurs when the TTR is leaning over on the stand?
If you decide to look at the needle valve be very careful removing the float retaining pin else you may need to so some tricky repairs
Brian's info is good, but before you do anything, check that there is no crud in the float needle valve. If flushing fuel through does not work, compressed air is your best friend. It is unlikely to need replacing but is something to keep in mind if nothing else works.
The float level would be the next thing to check, but unless you bent the float, it is unlikely to need adjusting. If you do, here is a simple check- connect the fuel line to see if it blocks the fuel off at the point shown below in the picture. It is the best way of knowing if this is correct. No fuel should be running or dripping out with the float level with the carb body, as shown below. You can also blow through the inlet with a long piece of pipe attached, but this tends to leave condensation in the inlet, so it's best flushed out with fuel afterwards if this method is used.
Actually even before you do that (because it's much easier, and might solve the problem), try tapping on the sides of the float bowl with a relatively weighty brass fitting or tool. Longer term, you may of course need to proceed with the more rigorous diagnostics and repair outlined below, but I find that this is a relatively common problem (sticky float), that is often resolved by simply tapping the float bowl with a dense metallic object.
About once a year I need to give a float bowl a few taps to stop petrol overflowing from a carb' one of my old bikes, but I feel it is safer to use a hammer handle or some other piece of wood.