One of the similarities is format: both begin with lectures on one or more topics which are of considerable length in comparison to the work and then follow this with an incident that exemplifies the topic(s) discussed in the first part. Poe's lecture is solely on the nature of perverseness in human behavior while Dostoyevsky's contains several themes, one of which is perverseness.
The following is a quote from Poe's "The Imp of the Perverse." He is speaking of perverseness when he says, "Through its promptings we act without comprehensible object; or, if this shall be understood as a contradiction in terms, we may so far modify the proposition as to say, that through its promptings we act, for the reason that we should not." In plain English, he states that we sometimes do things simply because we know that we shouldn't.
In Dostoyevsky's Notes from Underground, we find a similar theme. One of Dostoyevsky's first examples is that at times he is sick but doesn't go to the doctor out of spite. Who is he injuring--himself. He knows he should go because he is "only injuring [himself]...My liver is bad, well--- let it get worse." He is knowingly acting against his own best interests. Later he speaks of a "friend" of his:
"When he prepares for any undertaking this gentleman immediately explains to you, elegantly and clearly, exactly how he must act in accordance with the laws of reason and truth. What is more, he will talk to you with excitement and passion of the true normal interests of man; with irony he will upbraid the shortsighted fools who do not understand their own interests, nor the true significance of virtue; and within a quarter of of an hour, without any sudden outside provocation, but simply through something inside him which is stronger than all his interests, he will go off on quite a different tack--that is, act in direct opposition to what he has just been saying about himself, in opposition to the laws of reason, in opposition to his own advantage, in fact in opposition to everything..."
Dostoyevsky here, like Poe, argues that humans will act at times in direct conflict with what they know to be their best interests.
Dostoyevsky postulates an advance in science which might provide accurate prediction of human behavior while Poe points out a combination of phrenology and metaphysics that attempts do the same. Both then attack the possibility of a completely accurate science of predicting human behavior.
Dostoyevsky says, "science itself will teach man that he never really had any caprice or will of his own, and that he himself is something of the nature of a piano key or the stop of an organ, and that there are , besides, things called the laws of nature; so that everything he does is not done by his willing it, but is done of itself, by the laws of nature. consequently we have only to discover these laws of nature, and man will not longer have to answer for his actions and life will become exceedingly easy for him. All human actions will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms..."
And then, when complete rational harmony and prosperity is established, someone will stand up and say that we should "'kick over the whole show here and scatter rationalism to the winds' ... [and] he would be sure to find followers--such is the nature of man. And all that for the most foolish reason, which, one would think, was hardly worth mentioning; that is, that man everywhere and at all times, whoever he may be, has preferred to act as he chose and not in the least as his reason and advantage dictated."
Poe's narrator attacks this tendency to develop laws of behavior by pointing out several examples of when he and others went against their own best interests, both by acting and by delaying to act until it was too late.
Poe's story was published in 1845 while Dostoyevsky's came out in 1864. I don't know if Dostoyevsky had ever read Poe, so I can't say he was influenced by Poe.
What I also find intriguing about them is that both authors wrote "double" stories, in which the main character discovers there is someone else who looks just like him, who even has the same name, but who acts in a way that is completely the opposite. In addition, for some inexplicable reason, no one else notices the resemblance. This forces the reader to consider the possibility that the "double" is not a real person but an hallucination.
Dostoyevsky's story is titled "The Double," while Poe's double story is "William Wilson." I wonder if there is some relationship between these two themes--perverseness within the individual on the one hand and the doppelganger on the other.