We have air vents on the walls of our home. The top one is permanently open. Do we open or close the bottom one in the winter months, or vice versa?

Do they all supply warm air? If so I see no reason to shut any of them off. The idea is to get warm air into your living space,the more the better as long as they aren`t taking away from other parts of the home

The top ones may be your supply air and the bottom ones may be the return or visa-versa. With the furnace fan running hold a paper towel or a tissue up to the top vent and see it it blows away from the wall or is pulled to it. This will tell you which is which or that both my be supply vents. But if that is so you do have some return vents somewhere.

Open bottom in winter. It sounds like a old Coleman duct system from many years ago.

Back in your day hvac? Lol

Most likely your system and main trunks (ducts) are in the basement. I am assuming that these grills are directly above each other. If so, they are the return grills. The idea behind that is to close the bottom one during cooling to pull the warmer air from the top of the room, and during heating open the bottom one to pull the cooler air from the floor level. It sounded much better in theory than it usually worked in practice. The problem is that the low return in the winter caused you to feel the draft of the moving air if you were sitting close to the grill. In our systems that are fed from below I just close the lower ones and forget them. The air flow that causes the draft is above people's heads and they actually complain less about temperature that way.

Now attic trunked systems are a different story. I look at each of those systems to find what works best and do things differently depending on the situation. Different supply and return locations dictate different methods of operation.

Back in your day hvac? Lol

Yep