Replace or wash this furnace filter?
Replace.  Some say that filter media is meant to be washed and reused, but my experience is it passes ginormous amounts of dirt.

If it were up to me, I would throw that away and install 2" pleated MERV 7 (or 8) disposable pleated filters.  I usually purchase these from a local supply house that specializes in air filters because they will often sell any size needed for a reasonable price.  The prepackaged media available in hardware stores can be okay in a pinch, but they are often a poor value and those stores usually do not sell anything thicker than 1" pleated.

A 2" pleated Merv 7 Filter looks something like this:



Actual width and height are typically 1/2 inch under nominal.  Actual thickness is typically 1/4" under nominal.  (16 x 20 x 2 nominal is actually 15.5 x 19.5 x 1.75 to give installation clearance).  Air filter supply houses specializing in this media can usually fabricate these to order at a reasonable price.  These are called "pleated air filters, MERV 7 (or MERV 8)."  The MERV rating refers to the particle size that the filter traps.  Higher number gives tighter filtration, but most residential blowers cannot tolerate a filter more restrictive than MERV 8.