Have there ever been any islands that disappeared due to rising sea level?
Definitely, there are lots.

As you say, sea-level is rising in most (but not all) coastal areas. Indeed, at the end of the last ice age ca. 12,000 years ago, sea-levels rose about 120 m. Many islands and land bridges — such as Beringia across the Bering Strait, and Doggerland across the North Sea — drowned.

I can't find any lists of drowned islands, but here are some places you will find them today and in the geological past:


'Banks' on the eastern seaboard of North America would have been emergent during the ice age — Georges Bank and the Grand Banks for example. As they drowned, they would have formed islands — lots, in the case of Grand Bank.
The Hawaiian archipelago is part of a famous chain of drowned islands, but the change in sea-level is not the only thing that's drowning them. As the plate moves over an upwelling in the mantle — the same thing that's causing the volcanism there — it subsides and cools. I'm sure there are many drowned islands in other extinct and active volcanic provinces and seamount chains: the Zubairs, Canaries, Azores, Iceland, the South Pacific, etc. Sometimes life keeps up for a bit as a coral reef grows around the sinking island, perhaps forming an atoll.
Near-sea-level islands like atolls abound in places like the Maldives. These atolls are volcanic and actively subsiding. Sea-level may or may not inundate them, depending on whether the coral growth can keep up. Another fragile island is Sable Island off Nova Scotia — clinging to the edge of the continental shelf.
Volcanos can explode as well as sink slowly into the sea. Krakatau Island is probably the best example. This one has nothing to do with sea-level.
Rift basins start as mountainous tears in the crust, then subside and flood. Global sea-level change will often act to accelerate this. An example might be the various island groups on the shores of the Red Sea. These will likely sink into the sea eventually.
Salt-domes in Louisiana (e.g. Avery Island) form hills, but the domes continue offshore for hundreds of kilometres. Many of them would have been exposed in the past, some even more recently than the last ice age.


Maybe others can chip in some other scenarios; I'm sure there are dozens.

The point is that sea-level is not the only factor — subsidence, sediment accumulation rate, and erosion also play their rôles. But sea-level always plays some part, and can change quickly relative to the other factors.