The so-called gambler's fallacy is a fallacy if you are applying it to the outcomes of independent trials of some random process. An example would be a sequence of coin tosses. If you flip a coin for a long enough time, eventually you'll see a sequence of 5 tails in a row. Once you get to this sequence, it might seem as though heads is somehow "due", and that the probability of seeing another tails is something less than 1/2. This is a mistake, and it is what's called the gambler's fallacy.

But independence is a strong assumption that not all processes satisfy. If I'm drawing cards sequentially from a deck without replacement, you'd be correct to believe that after each successive draw, a card that hasn't been drawn yet is more likely to be drawn. And if after maybe 20 cards I still haven't drawn a club, you could even make an argument that "clubs are due", and start betting on clubs to come up next.

As for your question,

after a certain card has been played, is there a lower chance of that card being played again?

This process probably does not satisfy the independence assumption. The cards are played as a result of human decisions, and humans remember what happened in previous rounds. I don't know much about Hearthstone, but I do know that human decision-making doesn't look very much like independent trials at all—in fact, even when humans try to generate randomness, they tend to fail miserably at it.

So no, the Gambler's fallacy may not apply in general to card games like Hearthstone, or anything where the outcomes are generated by human decision making. With that said, it could go either way: maybe playing a particular card has been successful a few times in a row, and players gravitate towards that card.