Can I save a filtered view of my board?

Yes - for keeping a set of filters always active on your board, just use the Remember this filter option, listed in the filter menu:

It will remain applied for as long as you want, regardless of reloading the board view or closing the browser. In the latter event, when you re-open the board in a new window, you will be reminded that the filter is active with this little notice:

To remove the filter, use the Click here to clear filter option, shown on the filtering notice, or the Clear filter setting in the filters menu, or the x mark in the filters box.