No, screen sharing does not count against the total number of people you can have on screen or backstage on your StreamYard broadcast.
Free Plan
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.
Basic Plan
You can have 10 people in the stream plus a screen share on the Basic Plan, and up to 10 people can enter the broadcast studio.
Professional 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.