How important is game theory in Dominion?
One of the clearest examples of game theory in Dominion shows up quite frequently in 2 player endgames, and it's called the Penultimate Province Rule (PPR). Basically, you should avoid buying the second to last Province if buying the remaining Province would allow your opponent to win.

Imagine a game where my opponent and I each have 3 Provinces, with 2 left in the supply. My opponent also has 4 Estates, while I only have 3. If I buy the penultimate Province, my opponent will probably buy the last one on his turn and win by a point (27-28). Instead of doing that, it's better for me to buy a Duchy (even with $8 or more). Now my opponent is in the same situation I was just in - if he buys a Province, I can get the last one on my turn and win (30-28). So his best option is to get a Duchy as well. We'll both continue to avoid the second-to-last Province, doing a "Duchy dance" until somebody's deck falls apart from the extra green cards or the Duchies run out.

In this example, knowing what your opponent is likely to do, and is capable of doing, is extremely important. Sometimes it's necessary to break PPR and hope that your opponent isn't able to snag the last Province, and it's good if you can time it so that this is more likely - perhaps you were able to attack, or maybe the opponent just had to shuffle at an inopportune time, or maybe he just missed putting his Alchemists on top. If you have to break PPR, it's best to keep an eye on what your opponent is doing so you can find the best time to strike.