We certainly do have a responsibility to try to conserve resources so that future generations can have resources to use just as we do. This does not mean that we should stop using resources. Instead, it means that we should be sure not to waste them.
We do have a duty to our descendants. The Earth and its resources are, in a sense, a shared resource. It is a resource that we share with future generations. We do not have the right to use all of these resources or to use them without thought for the future. This is similar to how I have a personal responsibility not to waste my money so that I can have some to leave to my children (or to use to help them during my lifetime).
However, our generation does have the right to use resources. We do not need to deny ourselves the essentials of life in order to conserve resources for later generations. Instead, what we need to do is to try our best to find ways of having what we want while using as few resources as possible.