If all you have is a basin of water at 310K, you cannot concentrate the temperature. The water is already at thermal equilibrium, so there is no way to extract energy from it in any way. That's the nature of entropy.
Now if you have that basin of water at 310K and a heat sink at a lower temperature, such as a bunch of room temperature air at 300K, or the night sky at around 230K, you can start doing something. Now the system is not at thermal equilibrium, so there is the opportunity to extract work.
In such a case, you can indeed divide the water up into two buckets, one with a cup of water, and one with the remaining 999.9 or so liters, all at 310K. You can then create a heat engine (choose your favorite) which generates usable energy by transferring heat from the large body of water into the cooler heat sink. You can then use this energy to drive a resistive heater in the small cup of water to raise it up to a temperature worthy of brewing tea.
Your cup of tea needs to be raised in temperature by about 67K to reach boiling. If your teacup contains roughly 200mL, that means you have 999.8L of water to use as a heat source. If you transfer that heat into a 300K heat sink, that's a 10K drop. You can do some quick math to show that an engine that is just 0.1% efficient is more than enough to do the job.