Yes, what is now Anthony's Pier 4 is slated for development, and yes that development is starting soon, but no, Anthony's Pier 4 is not closing soon. The Globe reports that construction on a new complex at Pier 4 begins next month, but that the longtime restaurant doesn't have to budge until developers break ground on the park that will replace it, "which is not likely for several years." And so, like that odd, old history professor that has tenure but isn't retiring, it seems the restaurant will be sticking around for some time.
And even then, Anthony's could open shop elsewhere, as it previously said it would. A prior report has the restaurant, which opened in 1963, going into a hotel that's part of the new development. Which is a little bit like preserving that history professor's brain so that he can continue to lecture indefinitely into the future. For more details on the Pier 4 development, see Curbed.