I've never really tried washing plushies, but I have a few faux fur throw blankets.
I wash those on the gentle/delicates-cycle in my washing machine, and there's never been any issues, so I would imagine washing a plushie in a pillow case on a gentle cycle should be fine too.
The only thing that has happened to my blankets are that the fur has gotten slightly matted, but at this point, they're several years old, never brushed (because that would be silly) and in heavy use.
That's one of the biggest risks, but I know people who wash their fursuit bodies in the washer. They naturally brush and put in a lot more effort during the drying process, so those never seem to end up with any matting issues after washes.
The thing with plushies is also that they're stuffed and its going to take ages for them to dry. Throwing them into a dryer is a nope, as heat is quite likely to cause the fur fibers to warp and start matting.
But in short, I think the washer might be a safe option, for as long as you use a cold gentle cycle and give the plushie plenty of time to dry.
Hand-washing is a great and viable option too, especially if the plushie isn't particularly dirty, as you could just spot-clean or wipe the thing down with a wet washcloth (dunking the poor thing in a sink and scrubbing is quite likely to cause the fur to get matted) without getting it completely soaked. Again, it's important to keep in mind not to use too hot water. Colder is better.
Water mixed with some vinegar makes for a very mild bleaching detergent, so I'd recommend trying that. I've used such a mixture on my own fursuit without any issues.
Brushing it with a pet slicker brush as it dries should also help keep the fur nice and fluffy. (That goes for both methods.)
Personally, if one of my favourite plushies needed a wash, I'd go for the handwash cloth method, as I think that's the most gentle and least likely to damage it method.
Hope any of that helps. You might have already heard similar advice before, but I decided to write a slightly longer reply in case more people would happen to have questions about the same topic. ^^'