Will it hurt a tree if we let our Euonymus grow up it's trunk?

Its not great because it can cause structural problems, making the tree a bit unsafe.

Its not great for the tree- it can cause structural problems and make the tree unsafe

Euonymus spreads so rapidly it would not be advisable as eventually it will choke out the tree.

If your tree is mature, it can be beautiful. It comes in ground cover and climbing varieties. If you let it grow, you can control its direction and mass with simple pruning. It can be a stunning additon to the garden as a topiary/bonzai style treatment on a tree. It does not grow so fast that a couple trims a year cannot keep it nice. It is often seen on large mature estate and museum gardens.