No, screen sharing 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 screen share on the Basic Plan.
You can have 10 people in the stream plus a screen share on the Professional plan, and up to 12 people can enter the studio,, so you can swap people in and out if you’re doing a dynamic talk show.
You can have 6 people in the stream plus a screen share on the Free plan, and up to 10 people can enter the broadcast studio.