Is there an equation for jet engine efficiency as a function of turbine inlet temperature? I know that there is a relation between the turbine inlet temperature and the engine's efficiency. But, can anyone provide an equation or efficiency as a function of just the turbine inlet temperature?

I am looking for something like this: If the turbine inlet temperature goes up by 200K, efficiency goes up by 10%. Any numbers like this are appreciated.

Also, let's say that if GE invents a material that does not melt at all(just imagine), what is the highest temperature that can be produced by burning the fuel assuming nothing can melt.
The jet engine efficiency is unfortunately more complicated than just a one-to-one function between static Turbine Inlet Temperature and efficiency. Thermodynamic efficiency of a turbine engine is defined as the useful generated power extracted from the chemical energy added by the fuel. 

The following is extracted from a paper format uni book on aircraft gas turbines.



Station 0 is for the engine inlet, the other station numbers are:


Compressor inlet.
Combustion chamber inlet.
Turbine inlet.
Turbine outlet.
Engine exhaust.


Energy IN

The heat flow $\dot{Q}$ added to the engine is:

$$ \dot{Q} = \dot{m} \cdot c_{pg} \cdot (T_{3t} - T_{2t}) \tag{1}$$
With $\dot{m}$ = mass flow through engine, $c_{pg}$ = gas constant, $T_{3t}$ = total temperature at turbine inlet. Total temperature is the temperature reached when a gas flow is compressed isentropically, measured in the stagnation point, and defined as $$T_t = T + v^2/(2 * C_p) \tag{2}$$

So energy IN is a function of:


Static Turbine Inlet Temperature
total mass flow
gas flow velocity at the turbine inlet.


Useful power OUT

The power delivered by the gas generator is:

$$P_{gg} = \dot{}m \cdot c_{pg} \cdot T_{4t} \left[ 1 - {\left(\frac{p_0}{p_{4t}} \right)}^{\frac{\kappa_g - 1}{\kappa_g}} \right] \tag{3}$$

with 


$T_{4t}$ = stagnation temperature at  the turbine outlet.
$p_0$ = static pressure at engine inlet, a function of air density and airspeed.
$p_{4t}$ = stagnation pressure at the turbine outlet, which depends on how much energy the turbine has extracted from the gas flow.


Efficiency

If we could vary the $T_{3t}$ only and keep the other variables constant, we could indeed get to the sort of function sought after - but we cannot. Increasing T.I.T creates more thrust, accelerates the aircraft, increases inlet pressure, increases turbine outlet pressure etc.

From Gas Turbine Theory by Saravanamuttoo/Rogers/Cohen, page 81:


  The large number of variables involved make it impracticable to derive algebraic expressions for the specific output and efficiency of real cycles.