I recently bought a Canon EF-S 55-250mm F4-5.6 IS STM for my Canon T7i. This lens is significantly heavier than previous lenses that I've owned and I'm concerned with how I can safely store the camera with this lens on, as I'd prefer not to introduce dust inside the camera taking it on and off all the time. I typically store my camera in its bag on its back (with the fold out screen down, and the lens pointing upward) - will this damage the camera/lens if I store it this way with the heavier lens? Will it damage either even with a lighter lens? The other way it gets stored is just right side up, which causes it to lean forward very slightly and rest on the lens. Is this alright as well?
The Canon EF-S 55-250mm f/4-5.6 IS STM is a light-weight lens made mostly of plastic. There should be no harm to your camera from the weight if you store it with the lens facing upwards. However, the LCD may be scratched if you happen to rest it on any grit.
There is risk of scratching the lens if you place it face down in the bag without a lens cap. The lens in question isn't sturdy enough to rest face down on a shelf while attached to a camera. (You can try, but don't be surprised if it falls over.)
Resting the camera with lens attached so that the lens is sideways, at an angle, with the front in contact with a surface, is usually an inefficient use of space. Leaving it that way on a shelf for long periods of time risks distortion to the mount or barrel. This may cause zoom to not work smoothly or other problems.
Consider what happens to cue sticks left leaning against walls. Lenses usually do not have this problem because they are able to stand vertically without support. However, if you leave one at an angle for long enough, I'd expect gravity would win.
Some bags have partitions you can use to store your camera with lens attached. Whatever position your bag is designed for should be safe for your camera and lens.
Your EF-S 55-250mm weighs about twice as much as your EF-S 18-55mm. When compared to most other lenses, it is still a very light lens, and there is NO WAY it will damage your camera. Store it any position you find most convenient.
For me, the most convenient method is lens down in the camera bag. With the bag open I can grab the camera with one hand and be ready to shoot at a moments notice.
In the overall scheme of Canon's EF lens system, any of the EF-S 55-250mm variants are pretty light. The STM version only weighs about 375 grams, and all of that weight is at less than 120mm from the lens flange when the lens is stored at its shortest length. Your camera only weighs around 532 grams.
The same EF mount is designed to support cameras that weigh up to 1,530 grams (The EOS 1D X and 1D X Mark II) attached to lenses that weigh as much as 4,600 grams (EF 800mm f/5.6 L IS)¹. With lenses that are heavier/exert more force on the mount than the camera body does, we tend to support the lens and the mount supports the weight of the camera.
Storing your camera with the lens attached and the camera resting on its back should not hurt your camera if left in that position for years. Storing it sitting on the camera's floorplate with the lens probably tilted down slightly and resting on a shelf or pad in your bag should also do no damage to your camera or lens for very long periods of time.
¹ The discontinued EF 1200mm f/5.6 L IS was 16.5 kilograms, or 16,500 grams!