The answer is no under most conditions. And even if they could, it is unlikely that it would scare the fish.
First the physics. No matter how sensitive fish are to vibrations in the water, a fish investigating your line would have to sort the vibrations from your heartbeat from all the other vibrations in your line that come from other sources, including the line vibrating in response to the wind, the water, and your own movement. It's a matter of signal-to-noise ratio, and when you are fishing, the ambient noise in the environment outweighs the vibrations from your own heartbeat by several orders of magnitude.
Now the psychology. Even if the fish could pick out the rhythmic pulse of your heartbeat in the movement of your line, why would the fish know to be afraid of it? Vibrations occur naturally all of the time in nature, especially in water, most of the time for completely benign reasons, so it doesn't make evolutionary sense for fish to be afraid of all vibrations. The human heartbeat is relatively slow compared to other predators of fish, so in order for fish to be afraid of the human heartbeat, they would need to have evolved this fear directly because of human fishermen. The problem with this idea, though, is that human fishermen give off way more obvious clues to their presence that fish don't pick up on.
So no, you are not scaring away any fish with your own heartbeat.