anatomy of a tree

Many trees simply grow around objects like that, but it probably depends on the type of tree. At the very least I think we could say that it would not help the tree. But it might not kill it.

Of course it will influence it.
Likely weaken or kill it too.
Don't do that to trees, please.

When something restricts a tree as you described, it's referred to as girdling and can have significant effects on overall tree health. Girdling tends to harm and in many cases, kill trees by damaging the vascular cambium and restricting the translocation of photosynthates in secondary phloem.

In less severe cases, the tree may simply grow around the foreign object and may show little to no signs of deformation or reduction in overall health.

Well, In as much the two lateral meristem, the vascular and cork cambium which are responsible for the secondary growth (i.e thickening/increase in girth) of the wood and inner bark and outer bark respectively are not damaged by cutting, the tree will eventually overcome the wire stress with time and the wire may cut away.