There is some effect, certainly, but it is probably a poor strategy to depend too much on it. If you look at what the American Mathematical Society, for example, has published in the past three or so years you will get a sense of where the current action is. It won't include my old specialty, hard classical analysis, as the number of open problems there as decreased relative to those available in newer fields.
However, a given university will usually have a specific need when they put out an announcement. It might be a hot area or maybe not, since there is value in having a broad range of specialties on a faculty if it is possible. So, if I were 35 years younger, I might have a few openings available.
But there are two negatives that I can think of. First, things change. You are unlikely to be a serious contender for a faculty position for more than five years (I'm assuming you are asking about the US, actually). A lot can happen in a field in that time if those "essential to answer" questions actually get answered. The currently hot field may cool appreciably, to be replaced by the next big thing.
The other, and this is even more serious. It is hard to say where your mathematical insights will come. Mathematical insight is specific to a field (or even a small subfield), not general. You can have (as I did) tremendous insight into real analysis and almost none in abstract algebra (especially Ring Theory). I could solve the necessary problems in a course, but lacked the insight to push against the frontiers in algebra. You can seek insight and it may come, but it is an elusive thing. So, if you pick a hot field, a priori, just because it is flaming at the moment you may find the necessary insight impossible to attain while that area cools.
Rather, I suggest at your current level of development, that you seek a broad understanding of many mathematical areas and choose later, based on your insights and the then current evidence about the academic market, what you want to do. And the market is now very tight.
Finally, there is an old saying: You don't choose mathematics. Mathematics Chooses you.
But, I think the same can be said, almost as validly, for the various subfields within mathematics. I was "chosen" by classical real analysis.