Yes it is possible for any metal pipe to leach ions into freshwater supply in your home.
It depends on a lot of factors - but mostly how corrosive the freshwater is to begin with.
The lower the pH of the freshwater - the more aggressive it will be, and will corrode the copper pipe more than water with a neutral pH.
If you live in an older home with copper pipe, some or all of sweat end fittings could also contain lead solder.
Both copper and lead are bad news for marine tank owners - especially those that keep inverts - as I'm sure you know.
An RO or RO/DI unit will remove/reduce these copper and lead levels. How much is removed/reduced depends on the unit you purchase, the various stages of prefiltering, the quality of the RO membrane and the initial level of copper/lead in the tap water to begin with.
I've read where if you live in an older house with copper pipe w/lead solder fittings, that you should run the water for a while to flush the pipes out before drinking the water, or using it for consumption. I'd suppose flushing the pipes before using it for fish tank would be good idea too.
As the freshwater sits in the pipes, not moving, it tends to accumulate more of these metal ions, and by running the water for a few seconds/minutes, you flush many of them out of the piping system.
Many newer homes don't have this lead solder, but still have copper pipes.
Just another excellent reason for getting an RO or RO/DI unit in my opinion.
Or buy RO RO/DI water from another source.