welcome to GoY.
lift as much lawn up to about 3ft from the tree trunk. then use a weed-killer on the remaining grass. check the weed-killer is suitable to spray on woody stems without damaging the tree. When the grass is dead you can then put down the artificial turf. Make sure it has adequate drainage or that will impact on the water the tree gets.
If it was me personally I would stay with the natural grass. But situations change and that often means you have to make compromises.

I'd be a bit nervous as its a cherry- they tend to put out a lot of long surface roots, so they might only be about an inch or two below your lawn and extend over a fair distance after all those years. Laying artificial turf requires digging out soil too to put down granite dust or whatever you choose to use under the turf first, so its a bit of a risk...

You could always prepare yourself by taking the lawn up and seeing the situation - if you really want artificial turf and the tree dies, its an opportunity (or you can look at it that way) - maybe replace with a large, multi or single stem Amelanchier variety instead, which has blossom in spring and berries later on to ring the changes a bit, and which does not tend to put out those long surface roots.

Do remember that artificial grass needs shampooing and brushing so isn't entirely upkeep free and it does need replacing eventually..

Those cherry roots don't always stay beneath the surface -
sometimes they emerge and lie along the surface.