Acids become weaker in heavy water (Di-Deuterium oxide) than in regular water.
The pKa of an acid is a quantitative measure of the strength of an acid, i.e. how readily it gives up protons. pKa is related to the equilibrium constant Ka and Gibbs free energy change associated with the chemical reaction. This is all nicely described on Wikipedia.
The figure below [taken from this article] shows the isotope effect on pKa, i.e. the difference between the value of the pKa in heavy and regular water.
There appears to be a rough correlation with the magnitude of pKa, but for most Delta pKa ~ 0.5.
The largest value is for neat water (pKa=14).
So how is Delta pKa related to zero-point energy?
This way of looking at the problem is stated in a 50 year old J. Chem. Ed. paper by Kreevoy who says it allows students to see concrete effects of Heisenberg’s uncertainty principle.
It has the nice picture below, for acetic acid, which explains the basic physics. Dissociation of the acid lowers the zero-point energy of the proton/deuterium. Due to the H/D mass difference this energy change is less favourable for deuterium than hydrogen.
Why does dissociation [i.e. H/D removal] lower the zero-point energy?
Basically, the released H/D bonds to water to from H3O+ which will hydrogen bond with other waters to form units like the Zundel cation [H5O2+] in which the O-H stretch frequency becomes much softer.
Actually, doing a real calculation of this is rather non-trivial.
I am unaware of any attempts to theoretically produce the curve above showing the correlation between Delta pKa and pKa.
Update and correction (21 July, 2014).
Tom Markland kindly pointed out that if one considers a diverse families of compounds the correlation shown above between Delta pKa and pKa does not hold. The figure below is found on page 359 [Figure 11.4] of the book, Isotope Effects in the Chemical, Geological, and BioSciences.
The solid line shows a correlation that does hold for weak inorganic acids, including water.