What impact does heating or cooling have on the oxidation reduction potential of water?
I see two questions here.  The one that is stated (temperature effects on ORP) and a hidden one about water ionizers.

First, the temperature effect on ORP.  The thermodynamics of oxidation and reduction reactions is governed by the Nernst equation:
$$E(ORP) = E_o - \frac{2.3RT}{nF}\times log\frac{[reductant] }{[oxidant]}$$

In this equation, R and F are the gas law and Faraday's constants, respectively, $E_o$ is the standard reduction potential of the system of interest, T is temperature, n is the number of electrons transferred in the reaction and the concentrations of the reductant and oxidant are found in the log term.  The coefficient $\frac{2.3RT}{nF}$ in front of the log term can be simplified to $\frac{0.00198\times T}{n}$ and when we perform dimensional analysis on the term we find that it will provide units in Volts.  If we are looking for the largest effect that temperature would have on ORP then we want to use the smallest value of n, which would be 1 electron, further simplifying the coefficient to approximately $0.002T$.  T has to be in Kelvin so if we take our two extreme values as the freezing (273 K) and boiling (373 K) points of water, we find that the difference in ORP between these two extremes (all else being the same) would be about 200 millivolts.

Now to the hidden question about water ionizers.  There's a lot of hype about these systems with little scientific evidence to back up the claims.  There is at least one website useful for exploring the science behind water ionizers that is (I think) accessible to the general public.  A number of claims made by the manufacturers of these products, such as ion streams and water clusters simply don't stand up to scientific experimentation, and consumers need to be wary of the health benefit claims.