Asymmetrically loading barbell

You can happily do this with a leg press machine where the plate is solid with the mounts. It's one piece and if it was going to tip over, it would do it as soon as you took the weights off the storage pegs. It's one unit and does rely on symmetry. I do it all the time with myself and my clients without issue

The Smith machine can be similarly loaded, but many have a mechanism that assumes the bar stays level. It will work, but you could prematurely wear the track, or jam it and break the machine or hurt yourself.

If you're working with a free bar, it's advised that you don't.

On barbell, it's becomes a bit dangerous, you don't want to fail asymmetrically and drop the bar, it's a lot of weight. On a smith machine, a small weight variance is no big deal, go for it.

With dumbbells, yes you can assymmetrically load, it will greatly decrease your overall power output but increase isometric demand on abs, obliques, and spinal erectors to maintain stability. For example, some people like to do lunges carrying a dumbbell on only 1 side at a time for that kind of challenge.

Why would you bother doing this? What purpose are you trying to serve?

[off topic] I once heard a trainer suggest using less weight but doing the exercise on one foot.

If you're doing hammer curls it wouldn't make a difference because the weight is vertically oriented. With horizontal orientation, I wouldn't recommend because it would be off balance.