You need to read IRS publication 526.
https://www.irs.gov/forms-pubs/about-publication-526
There are two key points.
First, to claim any donation with a value more than $5000, you must have a signed appraisal, and you must get both the appraiser and an authorized representative of the donee organization to sign a paper copy of form 8283 documenting the donation. You mail the signed original to the IRS after e-filing the rest of your return.
Second,
Under the heading "capital gain property", you can generally claim the fair market value for donated property that would be subject to capital gains. However, there is an exception for "tangible personal property put to an unrelated use." In most cases, if you donate tangible personal property put to an unrelated use, you can only claim your original cost basis as the value of the donation.
Tangible personal property certainly includes coins, and an unrelated use includes having the organization sell the item to raise funds. In other words, if you donate the coin to a museum and they will keep it in their collection as a museum object, you can claim FMV (with a signed appraisal of course). But if you donate the item to the Salvation Army for them to auction to raise funds, you can only claim your cost basis.
If you are thinking of donating to an organization for an unrelated use such as fundraising, you would be better off selling the item yourself, paying the capital gains tax, and then donating some or all of the sales proceeds to the organization as money rather than as an object.
