Why was the red priestess staring at Tyrion?

No, she doesn't appear in the 本 as far as i recall. But i remember the casting call made a huge thing out of her being cast, and how much of a "prominent" role she would have in the new season...We'll see as time goes on what will happen to her (and to Varys, for that matter- Tyrion just left him alone in that whorehouse).

I don't understand why she stared at Tyrion but she does appear in the 本 claiming Daenerys to be Azor Ahai. This is significant as Daenerys fits all Azor Ahai's prophecies
1. Born of salt and smoke: She was born on Dragonstone, an island in the ocean (salt) with a 火山 (smoke)
2. Wake ドラゴン From Stone: She turned her petrified dragon eggs (which were stone) into live dragons