Why is the TPR valve dripping on my new water heater?
Sounds like you need to install an expansion tank, and possibly a pressure regulator (or pressure reducing valve).

When a check valve is installed, it becomes a closed plumbing system. Because it's closed the pressure in the system increases, as the heater heats the water. An expansion tank that is properly charged (charged to the "normal" water pressure) will "absorb" the pressure increase, keeping the system at a steady pressure. 

Follow the installation instructions provided by the expansion tank manufacturer. Typically the tank has to be on the cold water supply of the heater, and within a specific distance to the heater. You'll have to measure the pressure at this point in the system, to determine what to charge the tank to.

Expansion tanks typically have a maximum charge value, which should not be surpassed. If you measure the normal incoming water pressure (without the water heater running), and the pressure is above this rating. You'll have to install a pressure reducing valve (PRV), to prevent the incoming pressure from being too high. Plumbing code calls for a PRV, whenever the incoming pressure is greater than 80 psi (See this question for more info).