The new Logic Board will be as it was before you installed it. If the Logic Board was activated and functioning before it was transfered, it will still be activated.
iPhone activation is performed with the Original Telephone Carrier SIM Card. For the Original iPhone you need a functioning active SIM Card. For the other iPhones you just need an old Original Carrier SIM Card.
To activate iPhone without SIM cards, jailbreak iPhone and select the activation steps.