Do I play a hero's ability every time I draw it?
In Ascension you only ever use a hero's ability when you play the card. So in your case when you draw the Seer of the Forked Path nothing happens. However on your turn if you play that card you will draw an additional card and get the choice of banishing a card in the center row. The fact that the ability only happens when you chose to play the card is important because there are times when you do not want to activate a hero's ability because it has a drawback that you don't want to encounter.

If you look at the rules it describes what every area of the card does and for the area where the ability is it says:


  What the hero does when you play it


It should be noted that constructs unlike heroes stay in play and the effect happens every turn (though some of the abilities you have to chose to activate).

As a side note cards with the ability to draw other cards are generally pretty powerful and are good ones to look to acquire.