Filters that aren't designed to be washed should not be washed or vacuumed. For one thing, such actions may damage the filter, or if a damp filter is put back in place, it can encourage the growth of mold and the release of mold spores into your home's air.
There are washable filters available, although they are not common. Most are made of some sort of plastic. They have advantages and disadvantages.
Washable filters are more “environmentally friendly” than standard filters, since you might use one for up to five years, while your neighbor would dispose of about 60 non-washable filters in that same time span. The cost for a single washable filter versus multiple standard filters over the same time period is comparable. Maintenance is a little more difficult for a washable filter than for a non-washable filter, since you have to yank a washable filter out of your furnace periodically and wash it, as opposed to simply replacing it. But the real disadvantage of washable filters is that they just aren't very effective at trapping small particles.