For the stem - vice grips. I keep a small pair in my juice/tank/ancillaries toolbox now cause I don't have the grip strength to pull the post without bending/snapping something. Vice grips do the trick with very little force.
For the wick within the coil, you can dryburn without pulling that out, as long as it's silica (which, if it's a stock wick, it is) or mesh. Neither of those can burn - heat just cleans them. The reason you want to pull the flavor wicks before you dry burn isn't because you'll burn up the flavor wicks. It's so that you can see the coil, and so that the junk burning off the coil can actually burn *off* rather than just getting stuck in the flavor wicks/assembly.
Here's how I do it -
Pull the stem with some vice grips.
Remove the flavor wicks.
Screw coil/base onto PV.
(Check voltage, if you have variable voltage - I don't like doing this over about 3.5V)
Pulse fire the PV, slowly, until the smoke stops and you can see the whole coil light up when you fire it. If you've never done a dry burn before, it's worth noting that the first few times you fire it, it may seem like *nothing* happened. When you vape a lot of gunky juices, it can take several pulses for the gunk to start getting hot enough to loosen up and start to burn off.
As soon as there's no smoke and you can see the whole coil light, you're done. (you should also notice that that inner wick within the coil looks very clean.)
Blow through the coil assembly a few times to loosen any ash that might still be on the coils/wick. You can rinse it off, if you like, but then you have to allow time for the coils to dry, and I'm usually too impatient for that.
Cut new wicks (or burn the old ones clean with a lighter/torch... I rarely bother with that, cutting new wicks is easier).
If the new silica refuses to hang together after you've cut it, drip a few drops of PG or whatever flavor you're planning to vape next onto said wicks - it'll help the strands stay together.
Position the new wicks, trim any excess, and replace the stem.
That's it, you're done - Reassemble, fill, prime it, and vape.
You may get a slightly burnt taste at first if you don't rinse your coils after blowing out any ash, but it goes away within just a few vapes.