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It won't heat or cool the room, it's a heat pump so it will be neutral, pumping heat from inside back to inside

The heat pump uses electricity to move the heat around, and this electricity will eventually turn into additional heat which will end up in the room.

It will heat the room because the compressor is also inside the room, doing the work of moving the heat and thus generating heat itself.

Ok but I think waste heat in this context is fairly trivial, we might as well claim that the room isn't perfectly insulated or something so it won't actually get heated

It might be hotter outside than inside. Someone might think "It's hot, therefore I'll open my fridge to cool it down in here." But that will lead to it getting even hotter, not cooler. If the A/C is on, it'll require the A/C to work extra hard to remove that extra heat.

Also, would you claim a space heater doesn't heat the house if the house is at an equilibrium where the space heater produces heat at the same rate that heat escapes outside? I would say the space heater is still heating the house, but the house is losing heat at the same rate.

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