I've hypothesized for many years in my head that dark matter are the gravitational after effects of objects that have exploded. My understanding of physics is that when an object explodes, its center of mass remains in the same location. If that's the case, it seems that the gravity of that center of mass could remain. What observations or theories are there that counteract this hypothesis?
There are (at least) a couple of problems with your hypothesis.
The first problem is that dark matter is just that - dark, in other words it emits no light that we can detect. The exploded remnants of stars are not dark at all. Indeed the winds and superwinds associated with exploding stars are often extremely hot and luminous in the UV and X-ray region. Even when these winds have slowed and cooled, they will then form warm dust (because they are enriched in heavy elements) that is then detectable in the infrared part of the spectrum.
The second problem is that we know that the majority of dark matter is not just non-luminous "normal" matter. Estimates of the cosmic abundances of helium, deuterium and lithium produced in the big bang are sensitive to the overall density of this "normal" matter. These estimates tell us that only about 20% of the gravitating matter that appears to be required to explain the dynamics of the universe, the dynamics of galaxy clusters and the dynamics of galaxies, can be in the form of "normal" matter. But the atoms and particles released from an exploding star are just part of this "normal" matter budget - they have already been accounted for and so contribute nothing to solving this second dark matter problem.
Newton's shell theorem explicitly contradicts this hypothesis
Newton's shell theorem would seem to most directly comment on this hypothesis.
Historically, this theorem was instrumental in providing motivation to support Newton's theory of gravity, which at that time was a speculative model that physicists around Newton's time were considering following Hooke's model for springs.
A central premise of the shell theorem is that shells cancel out their own internal gravity. For example, a perfectly spherical planet would be expected to have zero gravity as its center. Also, Earth's gravity gets weaker as the center is approached.
So, say that a star explodes. Then according to Newton's law of gravity, the exploding shell should have zero gravitational pull within it.
This hypothesis would likely be modified Newtonian dynamics
For more information on alternative theories of gravity, Wikipedia's article on "Alternatives to general relativity" may be helpful.
In particular, this hypothesis would likely be described as modified Newtonian dynamics (MOND) since it explicitly contradicts Newtonian mechanics.