Make sure the NVIC is configured for flash interrupts. If using CubeMX, enable NVIC Flash intterupt and expect CubeMX to generate FLASH_IRQHandler() that calls HAL_FLASH_IRQHandler() in the xxx_it.c file. A breakpoint can be used to ensure that FLASH_IRQHandler() or HAL_FLASH_IRQHandler() are bing called.
You can define a HAL_FLASH_EndOfOperationCallback to perform some operation upon completion. Or, you can look at pFlash.ProcedureOnGoing == FLASH_PROC_NONE to know the operation is complete.
Note that if the purpose of using HAL_FLASH_Program_IT is to erase flash asynchronously while running from flash, it will not work as expected. Expect execution from flash to freeze while an erase is in-progress which effectively means HAL_FLASH_Program_IT() will block until complete. If you're executing from RAM, this blocking won't occur.