When I am in doubt I always turn to the Decalogue.
In this case I rely on Exodus 20:4-5...
So lets say I have one of those questionable coins..
1) Did I make it for myself? No
2) Do I bow to and serve this idol? No
Conclusion - Its not the idol.
At the same time, there is Deuteronomy 7:25-26. These verses do not require one to make it for themselves or to serve it. It refers to desiring the silver or gold, presumably implying one is not to even make any use of it whatsoever.
It says in the beginning of the chapter (Deut 7:1-5) that you would do it when you conquer enemy territory. Also, in Deut 7:25 it says "take for you"... so again... Do I desire gold and silver it is made from? No. Do I take it for myself? No. Yes, I do bring it into my house, but its not for a purpose of worship, neither these coins are the spoils of war. Plus I am in exile and I am a stranger in the land and its not my (and Torah/God's) land.
Deuteronomy 7 indeed refers to Canaan, but would one argue that an image from Canaan is an abomination but an image from somewhere else is not? V. 25-26 says these are abominations to YHWH, and not to bring such an abomination into the house. One is desiring the value that a coin has (similar to "delighting in the gold and silver") and keeping it/using it for that value. I would say this is prohibited by that passage, and that the passage is speaking further than just in regard to worship.
I think the problem lies in the definition of the "graven" and "molten" images. I use the 10 commandments as a reference point because they were written in stone and (as I understand it) is a bare minimum required for observance. Verses of Deuteronomy simply expand on the idea. However, everywhere I looked it appeared that "an image/representation" is only considered an abomination when it's being bowed down to and served to. I do not think Torah implies that all works of art (particularly stone sculptures and molten sculptures) are in any way an abomination.
If we would have a Christian god on the coin, then it would be clear without any reservations. However, since we are talking about some obscure Roman deity that no one really worship (or bows down to) here in US, I do not think these Torah verses can be applied here. Just because of a few neo-pagan wackos that happen to worship it (which I doubt), does not mean that it is a deity.
To be honest I see this problem as a gray area. I am not sure if I am correct.
There are Numb 21:8-9 and 2 Kings 18:4?
Carvings/castings of forms in general are not what are prohibited. The example of the serpent is a good demonstration. The issue is whether it is deemed "elohim". The serpent was not initially that, but it soon was treated as a deity, and so it was seemingly rightfully destroyed according to 2 Kings.
It seems that a being (specifically an image of that being) that was once worshiped can achieve a point in time where it is no longer considered a deity, assuming the worship of it is no longer popular. This is incorrect perspective for a number of reasons. For one thing, if something was created as an image of a deity, it would inherently be a "carving" or "casting", from my understanding. If the people become apathetic and no longer worship it, I see no precedent for it to no longer be treated as such. By its very definition it was an image of an elohim, whether the people believe in it or not (that god isn't real regardless).
The most obvious idol cast into American coinage would seem to be a very real object of worship, even if people do not "know" that it is a Roman god or "worship" it in the sense of burning incense/etc. If I recall correctly, its name is printed on every coin minted in the US and it has multiple images that are cast into coins which are not circulated (such as precious metals). In CA, it may not be treated this way as much, but I remember people holding that image to be very important especially back around 9/11.
Ultimately, from my understanding, these images were originally created to be images of elohim, therefore they would be idols, even if people decide to not worship them and just treat them as "symbolic" images (which ultimately would connect with an elohim idea anyway).
So destroy/dispose of the New York quarters. I also strongly question the Pennsylvania and New Mexico quarters...Pennsylvania has a woman statue on it that seems deity-like in its description, so I strongly question it, and NM has a religious sun symbol but I couldn't find much information on whether it was actually worshiped. So I'd destroy/dispose of them.
Then what I do to keep from getting more is to ask for dimes back instead of quarters at the store. "Could you give me dimes back?" I check my quarters if I get change back at a store with them and then just leave/exchange them if they're one of the problem ones. You can also use debit/credit cards so you do not have to handle questionable objects.
Concerning presidents, etc, I wouldn't think they would be elohim in that sense. I wouldn't think actual paper bills would be any issue since they're not carvings/castings.
There is also Connecticut quarter with a tree on it (Charter Oak), that can be considered a carving of "asherim" (Deut 7:5). But I don't think this one would be a problem, since I didn't see any indications of worshiping it or treating it as any sort of being (let alone supernatural) and it would not seem to be a carving either.