What's the non-poking action that makes Partner Eevee sad?
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This video's https://youtu.be/g4nUzEvlpGA?t=551 shows that doing a certain action makes Eevee sad, and it's not poking, because poking angers, not saddens.
What is that action?
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I think this is "petting":
In Pokémon-Amie and Pokémon Refresh, the player can pet their Pokémon by using the touch screen. Petting a Pokémon increases its affection. Most Pokémon that can be petted have places where they especially like being pet and places where they dislike it. https://www.google.com/url?sa=t&source=web&rct=j&url=https://bulbapedia.bulbagarden.net/wiki/Petting&ved=2ahUKEwik5trt2ZP4AhXS4HMBHYfSBrUQFnoECAgQBQ&usg=AOvVaw3fdOO2qBn_8pPJg6Jinxn3
So I think that petting it too much or petting them in places where they dislike it can cause it to be sad.