Can grass killer be used by a tree??

I have been thinking about this. I don't think that using a grass killer would harm the tree, but it depends on two things. 1. How big and well-established the tree is. 2. What kind of weed killer you intend to use. I would avoid those with glyphosate in. Read the label before you do the deed! Then repeat if the grass returns. Dig out all remains of roots afterwards.

Another consideration is how close to the surface the tree roots grow. If you can see tree roots through the grass, or hit them with a spade, any grass/weed killer would directly contact the roots.

Thank you for your help, the tree is a well established and some of the roots that are close to the tree are just under the grass and this is the part we wanted to put grass killer down, is there anything else we could do?

A fast way to kill grass in the summer is to cover it with black plastic for a few days. Without sun or water the grass simply rots in the damp darkness. For a new bed in open area, you can actually allow sun to "bake" soil a bit - most weed seeds will die. (passive solar solution) Under your tree you would not want to do this very long! But a well established tree will get water from deeper roots.

Barrierisland has given you good advice...I have seen the results of accidental spillage of 2-4D... it was not pretty... the conifer grew in all directions from every growth point in a corkscrew fashion...grass killer or any herbicide works by stimulating the plant with growth hormones...it grows itself to collapse and death.
Solarization is a much better method. just water your tree well before putting down the plastic.

Thanks for all your help :)