In general, yes, but how severely depends on the species and the context. Some species exist in a narrow range, and are not either the sole predator for any other species, nor the sole prey for any other species. If species like these disappear, then other species simply move in to eat what they would have eaten, and the things which would have eaten them simply turn to a new food source -- although it may mean that the "new food source" is then impacted more. There also may be subtle effects, like perhaps the balance of plants in the area alters because the things now eating those plants graze in different patterns. These things cause ripples, but in general everything adapts. If a single beetle species in a rainforest disappeared, other species might move in to take its place with scarcely a blip, but it depends on what that original species did and how it interacted with what was around it, and how similar the species taking its place might be. On the other hand, some species are key links that the local food web can't do without; they may pollinate trees or flowers that no other species pollinates, or they may be a key food source for a variety of other species themselves. An example of the former would be a single species of pollinator wasps which pollinate the tree Ficus rubiginosa, which in turn produces fruit eaten by a large number of birds and animals. If those wasps disappeared, the trees would no longer fruit, and not only would the trees eventually disappear in the long run, but many other species would be deprived of a food source in the short run. An example of the latter would be sandeels; an unpreposessing little fish which feeds not only larger fish, but a wide variety of seabirds as well. In areas sandeels have been fished out, because they are ground up into meal to feed farmed fish like salmon, and because they are made into fishmeal fertilizer for gardeners. Where sandeel have disappeared there have been massive seabird die-offs, as entire generations of chicks have starved, and the larger fish have been affected as well, which then affects the seal populations. In short, the answer is yes, but it's not the same for everything.

Absolutely, we are all interdependent on one another. The affect might be or seem relatively minor, but all such changes produce dislocations. Let's take the example of a plant becoming extinct. It probably provided food for certain animals and / or insects. If one of them was completely dependent on the plant as a food source, they also would become extinct unless they could adapt quickly. The animals that ate the plant also become the food of other animals further up the food chain. The decaying plant material provides food for smaller organisms, microbes, yeasts, molds, etc. Many of these are highly specialized and are directly affected when their food source is eliminated. The plant may have had a symbiotic relationship with another organism. This process occurs constantly around us as a natural part of life. However, external influences can produce significant dislocations - mankind being one of those external influences. When we are responsible for the termination of a species, through pollution, overhunting, habitat loss, or any other reason, we diminish the world in which we live.

in short: of course it removes a link in trhe food chain.