Let's say that, since the dawn of time, any time an organism has died, it has dissipated upon death. With no decaying matter to fertilize or produce future plants/animals, would life have adapted? Would animals have evolved to be non-carnivorous? Would plants be non-coniferous?
Life would exist very briefly, before exhausting every resource
The recycling of elements is vital to how our ecosystem evolved into existence. Whenever an organism dies, detritovores return the nitrogen in the organism back to the soil.
But if this nitrogen simply disappears with every death, it goes nowhere. It doesn't get returned back into the soil, and as a result the autotrophs would die out quickly after using it all up in the first era of the evolution of life.
Life would exist, but not long enough for any complex life to evolve.
Sydney is exactly right - if you're destroying matter, life will burn itself out wherever it starts. However, let's say you tweak things a bit. Maybe instead of being destroyed or decaying, it is reintroduced into the environment by other means. Perhaps this world is a simulation or experiment, where the material is "magically" removed upon death, but then added back in with rainfall. Perhaps there is creation of matter occurring by some other means to roughly balance the destruction of matter. I'm also going to assume that destruction of matter occurs on a cellular level, otherwise there are many more problems that make life impossible. In this case:
We're modifying the Lion King's "Circle of Life." All creatures must have a way to get the resources that, in our world, they get from previous deaths. In our world, all living cells are either Autotrophic or Heterotrophic. Autotrophic cells generate energy from processes like photosynthesis, while heterotrophic cells rely on other organisms, via eating, for example.
In your world, all heterotrophic cells (or the organisms composed of them) have some difficulties. Eating a living thing only works if the cells are not killed in the process. So rather than eating a plant and converting it into energy, your heterotrophs would have to eat it and "annex" its cells. The eaten organism can "die" in this process, but its cells must not, or they would disappear. A heterotroph would need to replace its dying, energy-less, cells with the energy-full cells of another organism. Since nothing on earth does this, (at least not that I'm aware of) anything past this point on your heterotrophs would be pure speculation, and you can invent them as you wish.
Edit to add more info about Heterotrophs: I'm not a biologist, so I'm wildly speculating, but I imagine this "annexation" process as being sort of like getting a transplant on a cellular level. If a person kills their liver with over-use of alcohol, they can have it replaced with another person's liver. I'm imagining a creature that, instead of supplying it's own cells with energy to keep them alive, replaces them with another creature's cells. Simple organisms that use this technique would have to be built of the same cells as their prey, but maybe more complex organisms could evolve a method of modifying the cells they "annex" to suit their purposes. As for how the cells are "annexed," I was imagining a creature that ate it's prey, but rather than using a digestive system like ours, has a system specifically for sorting out the cells it finds useful and moving them to the proper place in its body. That process would be very complicated, and it's beyond my knowledge level to even say for sure that it's possible. Alternatively, you could make something really freaky that transplants whole portions of its body. It would need redundancy of all vital systems so that when one died, it could live on. I'm imagining a sort of plant monster that builds itself out of mobile autographs by grafting them onto itself. It probably wouldn't have any kind of consciousness recognizable by humans, as I don't see how it could manage transplanting brains without changing its identity entirely. Either way, I think the result would be fairly horrific, as it would need to be "feeding" constantly. Cells don't live very long without a supply of energy. End Edit
Autotrophs have fewer problems, but they're going to look and behave differently from our autotrophs. On earth, most complex autotrophs are stationary, more or less. (Think plants) In your world, they can't rely on death to bring them things like nitrogen, so (assuming matter is not delivered to them by some magical or external process) they either need to move to seek out their nutrients, or, if the created matter is distributed fairly uniformly, they will need to reproduce fairly quickly and in a specific direction. A plant-like thing could exist, but it would have to always bear offspring in the same direction and at a significant enough distance that the offspring was not relying on the same patch of nutrients as the parent. The creature can't use any kind of north/south direction for this, or it would die out when it reached the pole. It needs to use a direction relative to itself. In this way, it can use up the matter at its location, then bear offspring in the next spot over. Presumably your matter-creation system will have renewed the nutrients at the original spot by the time it's offspring have circumnavigated the planet via reproduction. (Or maybe the direction is a little more random than that so we can avoid hostile terrain and have multiple offspring - that would work better, but still, some descendant will eventually occupy a previously used spot, and that spot must have its nutrients renewed by then) Aside from something like that, you're looking at autotrophic animals. The only ones on Earth that I know of are some sea slugs and coral. Coral doesn't do much for us, since it functions more like a plant for our purposes. You want something like sea slugs. It needs to use a process like photosynthesis or chemosynthesis and be able to move to find the nutrients it needs for that process.
I can't imagine any other sort of life without allowing immortality, and I don't think that's what you want.
In this scenario do you consider plant life to be living things? If not, then maybe, MAYBE, all your creatures could be vegetarian and the plants would survive by living off of their own "circle of life".
In other words there would be the plant cycle and your "living" creatures (people?) are basically a parasite off of the plants "circle of life".
Otherwise I don't see how life can continue to exist without the exchange, if you will, of energy. If a creature disappears as soon as I kill it how can I eat it?
p.s. is this a side effect of the infinity gauntlet? lol