Does the new virus tree of life change their position between living and non-living things?
The definition of life is a question of philosophy

The definition of life is a question of philosophy and not of biology. However, because discussing it often require some knowledge of biology and because it is so common for the general public to misunderstand that the definition of life is not of biological interest, the question has already been addressed on this biology.SE. Please have a look at:

why-are-viruses-considered-microbes and why-isnt-a-virus-alive.

Why would the tree of viruses affect our definition of life?

It is impossible to say much without a link to the original claim. I will need to speculate on what the authors of this claim first meant.

A tree of viruses gives a visual representation of a feature that we tend to associate to living things only, that is they evolve and speciate. Of course, we did not need to build a tree of viruses to understand that they evolve and speciate but a visual representation can sometimes offer an emotional response that will push some to prefer considering viruses as living things. In consequences, it is possible that the vision of this tree affect the emotional aspect of what we define as "living things" and might push the community to expand their standard definition to include viruses. Again, it is important to understand that the definition of life really has absolutely no impact on Biology.