Interesting question. The Flamingo's commemorative bricks are novelty souvenirs that enable you to leave your mark on a little piece of Las Vegas – a 4" x 8" plot in the hotel's tropical garden area, to be precise. The bricks can be engraved with up to three lines of text, 20 characters per line (including punctuation and spaces), celebrating a special event or in dedication to a loved one, for example. According to the official order form, which you can download from their site by clicking here, it takes approximately three months from the time you pay for the engraving and installation process to be completed and at $150 each, they're not cheap. No wonder you're concerned about the fate of your brick.
We called the retail department of Harrah's, which of course now owns the Flamingo, and posed your question. Not surprisingly, the official word was that there are no plans in the pipeline to close the Flamingo, so the query is moot. We said OK, let's just suppose — purely hypothetically — that somewhere down the line the Flamingo is imploded. What would happen to the brick then? Could it be saved?
The response was that the bricks aren't fixed in concrete, but merely nestled into some sand, and thus, in theory, are retrievable. Seeing as you have purchased your brick outright for a one-off payment, we figure you're the undisputed rightful owner. So if you ever hear of plans regarding the property's demise, you should step up and retrieve your brick before it's buried under the rubble of an imploded 4,000-plus-room hotel.
