No, you can use it without purchasing a new one.

The license file contains the version and expiration date. If you update the license file from the web, the expiration will be extended for 15 months and you will be able to use ModelSim-AE.
However, please note that ModelSim-AE versions released after the date of version cannot be used.

For details on the contents of the license, please search for the keyword "FEATURE and INCREMENT Lines" in the application note at the URL link below.
For the license expiration date, search for "Specify the License for the ModelSim-Altera" from the manual at the URL link below and refer to it.

  https://www.altera.com/en_US/pdfs/literature/manual/quartus_install.pdf