My understanding was that it gets cut into pieces (probably nucleotides in the end) and then reused. I've done a quick google search and it seems that there are multiple mechanisms: http://www.eb.tuebingen.mpg.de/remco-sprangers/mrna-degradat... and also https://en.wikipedia.org/wiki/Messenger_RNA#Degradation

But the main idea is that it degrades with use until it becomes unusable and actively gets destroyed.