No, screensharing does not count against the total number of people you can have on screen or backstage on your StreamYard broadcast.
You can have 10 people in the stream plus a screenshare on the Basic and Professional plan.
You can have 6 people in the stream plus a screenshare on the Free plan, but up to 10 people can enter the broadcast studio, so you can swap people in and out if you’re doing a dynamic talk show.