Answer: Rambam (Shabbos 25:6) writes that as coins cannot be used on Shabbos, they are muktze just like raw wood and stones that have not been fashioned into anything. The Mishna Berura (310:24) writes that they are muktze machmas gufo, inherently muktze.
While the Shulchan Aruch (OC 308:21) writes that rocks are usually muktze, this is only when they do not have a specific use. If one prepares a rock for a specific permitted use before Shabbos such as to keep a door open or to crack nuts, it would no longer be muktze. Thus, the Mishna Berura (303:74) writes that if one set a coin aside for a particular purpose it is considered to be non-muktze. This only applies when one does so in a permanent manner, such as creating jewellery out of a coin, but not if one sets it aside for just one Shabbos (See Shemiras Shabbos Kehilchasa 20:38).
The Chazon Ish (Shabbos 42:13), disagrees, writing that one cannot set a coin aside, as one may choose to use it again as money. Thus, coins will remain muktze.
R’ Asher Weiss (Minchas Asher 2:36) writes that when one sorts such coins into a collection, one is treating them like a photo album whose purpose is simply to be viewed at one’s leisure. Nonetheless, he questions whether it is ideal to be stringent and avoid touching them on Shabbos.
R’ Yisroel Dovid Harfenes (Nishmas Shabbos 3:331) argues, however, that even the Chazon Ish would agree that a coin collection is not muktze. One who goes to the bother to collect various coins of different denominations and time periods is certainly not going to break up their collection to spend the money. Such coins would not, therefore, be muktze.
In conclusion, a coin collection is not muktze and may be handled on Shabbos, especially if the coins are no longer in circulation.