Is there any merit to using tap water over filtered water for making ice cubes?

Absolutely, hopefully your filtration water screens many copper filaments that fester and accumulate as the water comes through the plumbing pipes… I’d stay with drinking water from the store… the chlorine content in city water can present a serious health risk.
The turbidity in your water is important to have an appreciation for. HTH
http://www.jeremyanderson.com/blog/2006/07/07/package-design/

The chlorine content of public water systems is not a health risk to humans, frozen or not.

Tap water in America is safer and held to higher standards than bottled water. Filtered water is only any better if you maintain the system and change the filter as frequently as required. Otherwise, you can end up with more dangerous water.

As to the original question, I had never heard that “old wives tale” and see you no reason why it would be true.

Most municipalities don’t put Fluoride in thier water any longer. That is how they save money. Where I live every bit of water that inside my body and cook with is filtered.

Where I used to live in West Texas there is a huge difference. Ice cubes from tap water come out “fuzzy” and very cloudy almost opaque. Filtered water or in my case Reverse Osmosis water made crystal clear very hard and smooth ice cubes. The big deal for me was that cubes made from filtered water melted half as fast as those from tap water. I never figured out why, but I suspect it had something to do with the mineral content of the tap water causing the ice to be less dense.

Other than the nasty taste of tap water??? I can’t even use ice cubes from the tap or fridge (if not filtered) because I can taste the minerals/chemicals in whatever I am drinking…yuck!