Not necessarily. HED has a diagram (which I can't find at the moment) that shows actual profiles for a GP4000 on 19, 23, and 25mm (external) width rims. Going from 19 to 23 on both 23 and 25mm tires resulted in a significant height increase. Going from 23 to 25mm width rims, the height of the same tires remained the same.
In a simplified "theoretical" model, the tire assumes a circular shape. Adding width to the rim is effectively increasing the tire cross-section circumference. In this model, the tire diameter will increase by (rim width increase)/pi. So a 2mm width change would be a 0.64mm increase in both width and height. However, real tires aren't uniform in cross-section and don't assume a perfectly circular shape. Under pressure the casings tend to flatten at the sidewall so more of the rim width increase goes into increased tire height than width (up to a point).
tl;dr; your tires probably won't get shorter on wider rims and may even get taller. If you're tight on clearance, make sure you can test fit before committing to purchase.