I'd say it's almost certainly load bearing for one simple reason.
You've got what appears to be a beam running over the opening from the dining room to the hallway, and you don't have one going into the kitchen.
The fact that your dining/hall opening isn't floor to ceiling suggests strongly that the beam is running across that opening, which means it's running the depth of the house.
It can be kind of hard to tell from photos.
Your house plans (blueprints) would tell you for sure, presuming the house was built faithfully to the plans. You really should have a good look at the plans, or get somebody knowledgeable to look at this in person. Or both.
Do you know which direction the joists are running? Floor and ceiling joists will be perpendicular to the load bearing walls (or beams!), not parallel to them. If the joists pass across the top of the wall (or beam) unbroken, then the wall is possibly not load-bearing, depending on your joists, the length of the span after you remove the wall, and the load on the floor above.
If the ends of the joists are resting on the wall or beam, then it is definitely load-bearing.
if there isn't another floor above this one, and you have engineered, prefabricated trusses holding up your roof, this won't be load-bearing. The trusses are engineered to handle the load of the roof by themselves. If they aren't engineered trusses, that doesn't apply.
Is the long common wall of the living room/dining room/kitchen directly on top of (parallel with) the long wall downstairs, and is the opening between the dining and living rooms perpendicular to that long wall downstairs? It looks like the photos might be from different directions? If so, then the common wall with the kitchen is probably the load-bearing wall, and the opening between the living and dining rooms may or may not be, depending on how the joists are laid out.
But you really should look at the plans or get somebody knowledgeable to take a look in person.
Best Answer
The only thing bearing on that wall are shampoo bottles. Knock away. The wall is empty. Note that the end of the porcelain tub is open, so something will have to go back to cover it up. When it's open you can inspect under the tub for horrors.
But you'll also want to remove the wall sheathing all around that tub, and read up about flashing and backer board.