Does simmering sparkling water change/decrease its mineral composition significantly? I know that heating sparkling water will cause it to lose $\ce{CO2}$.

Does this have any other effects on the water that remains, such as reducing its mineral content?  I understand when non-carbonated mineral water evaporates, its minerals stay behind and create a greater mineral concentration in the water that's left. Does sparkling water behave in the same way, or do other reactions occur that cause the water to change in additional ways?
No, simmering sparkling water should have negligible effect on its mineral composition.

You're exactly right that simmering it will drive off the $\ce{CO2}$. This $\ce{CO2}$ leaves as only/exactly $\ce{CO2}$ molecules, and any minerals that might have been associated with the carbonate/bicarbonate $(\ce{CO3^{2-}}/\ce{HCO3^-})$ will remain behind in the water.  Most such minerals will be present in dilute enough of concentrations that they will remain dissolved in the no-longer-sparkling water, as long as only a relatively small amount of the water itself is boiled away.