Quoted from LTG:

Have you tried removing the solder on the back of the board. If you get it all they should pull up.
Otherwise pry the plastic up with a thin screwdriver. Start at one end. And wiggle it up. If you are replacing the plastic and pins. Cut the pins off close to the plastic, less pin for the plastic to go up then.
LTG : )

Quoted from vid1900:

It can be tough to remove without a desoldering station if there is a lot of solder balled up on the back side of those headers under the plastic.
Using a brand new blade in your razor knife, slit the plastic lengthwise along the pins, in one long cut. Don't get too excited, cut about halfway through.
Now flex the resulting plastic tab slightly, while you make a second finishing lengthwise pass
Wiggle the other half still attached to the pins and it will come right off.

Quoted from sharkey1331:

I've had good luck with first re-solder, then de-solder, and then use a hot air rework station to heat the pins, and when the remaining solder is hot enough, they pull out cleanly all in one (the whole connector group). This has worked the best for me, rather than try to remove the white part, or cutting the white part so you can remove each pin individually.
Without hot air, my second string attach would be, carefully cutting the plastic into seperate pins, and re-solder, then heat the solder while pulling out the pin, then de-solder.