Instant open piston door with delayed deactivation with pressure plates

Solution 1:

A simple pulse lengthener will work here.

It works like this:

The first block gets powered from the pressure plate (or you can power it in some other way), the redstone line immediately takes power from there, so all the pistons go up instantly. After some time the repeater draws power from the block and powers the next block, and so on...

When you step off the pressure plate, the first block no longer powers the redstone line, but all the next blocks still do; the first repeater turns off after some time and the 2nd block is not powered anymore... but the line is powered until the last block loses power.

If the signal is shorter than 4 ticks, this scheme won't work well, but buttons always give a 5-tick pulse and pressure plates' signal lasts at least 5 ticks.

And by the way, I used iron blocks, but these can be any non-transparent blocks,

Old answer

Or, for a different behavior, use an RS NOR Latch as in this screenshot.
This scheme will keep the pistons up for at least the time it takes the current to go through the repeaters, or for more time if the player keeps standing on the pressure plate.

Solution 2:

You can place a line of repeaters in parallel with a straight redstone wire and adjust the repeaters to give you the desired off delay.

The only catch is you have to let the repeaters "charge up" to get the off delay to work. Otherwise, the piston will turn off, the turn on, then have a delayed off.