It will not hurt to leave the violin case open, but I do suggest using a silk scarf to cover the violin and loosen the bow while not in use. The scarf will keep dust and little unknown living things off your bow and violin and keep your case clean.
Try to keep the case off the floor. Even if it is kept in a plastic tub that sits on the floor, that is better than sitting on the floor. Leaving the case on the floor invites dust mites to get in and start feeding on your bow hair. You won’t see the bugs, but you’ll notice lots of bow hairs mysteriously broken. This is a signal of dust mites and other bow hair eating critters.
Also, keep your violin away from central heating and air conditioning vents or any sources of heat or cold. Even a window that allows hot, full sun during summer is not an ideal place to allow your violin case to sit. Especially never leave your violin in the car in extreme temperatures, whether hot or cold. Changes in temps cause the violin to go out of tune but worse, consistent exposure like that can cause the violin to suffer opening seams. You’ll notice a buzzing or vibration on one specific note or a range of notes and after checking all the usual causes and finding everything intact, you still have weird buzzing. That could be an open seam or the sound board has come loose, inside the violin.
Here’s a pic of the sound board inside a violin:
Also try to keep an eye on the humidity of the environment where you store your violin. High humidity is ok, but low humidity can cause the wood to contract to such a degree that seams open, too. These seams are glued by the maker and the higher quality your violin is the more likely it is to respond to changes in temp and humidity. In dry areas, it is useful to use a DampIt to maintain an ideal level of humidity. If you love your violin, you must keep a DampIt on hand.