So we finished our first game however one small rule (probably more we didn't notice at the time) got us a bit muddled.

Some objectives has a special abilitiy, the one in question 'Forgotten Heroes' was on the table, my opponent used the resource there and a few others (now has a focus token) then played Obi-Wan Kenobi. Would the objectives special ability letting them draw a card still apply considering that the card is technically exhausted.

Yes it still works only abilities that would not work are those that require the card to focus. This is in the rules on page 27.