It's been said before, but I'll skim through a few things.
Kano Kroil is far and away the best penetrating oil I've ever used. It can penetrate to less than 50 millionths of an inch. (.00005'). The stores won't have it, you need to buy it from a dealer, or direct from the Mfg.
It also has a way of eating away at rust, and not letting it re-form. Some oils will just break the rust loose and re-distribute it. This stuff dissolves it. Not like an OA bath will, but the objective is to get it to move, not remove the rust yet anyway, but some dissolving is better than no dissolving here.
They call it
the oil that creeps, and that it does. So you have to let it soak. The deeper it gets, the less friction you have.
The rust is kind of like stalactites and stalagmites, eventually they come together and become one in some areas. Surrounding them, and soaking them in Kroil helps a LOT to soften them up.
There are a lot of variables. Some seat posts, were made of nothing more than cheap seamed tubing, and dirt materials. They are very soft, and it tears, and swells, and bends very easily, and can be virtually impossible to get those out.
Others, are made from quality tubing, like 4130 cro-mo, and were chrome plated and such. With these you have a much better shot at getting then out, and are worth spending more time trying to get them out before cutting. Kroil will usually do the trick on these types.
Aluminum alloys are an odd duck too. But Kroil will soak right into the aluminum oxides too, and at the very least provide deep down lubrication.
What type of material the frame is made of matters a lot too. It's all a big variable, not a one size fits all solution with seat posts.
So always start by soaking it with the best penetrating oil you can find. Hoppe's makes a good one, and some others. Soak them inside and out, upside down, and upside right.
Once they are soaked for hours and/or days you can start working it one step at a time.
The reality is, that you can put more force into the seat post by pushing it in, than you can pulling it out. Even with a good slide hammer, or porta-power pulling on it. This is because you can set it on the floor (hard rubber mat), and hit it with a huge hammer, and the force all transfers to the post, and it's backed up solid. In fact, you have to be careful not to dent the BB, because you can deliver that much force hitting it.
I've said it before, and I'll say it again, ...hear me now, or hear me later, lol

... The magic of a good penetrating oil is two-fold (or more). First, it can get into REALLY small places. Second, it will displace water. What this means is, that when you start hitting, pulling, and everything else, the droplets of penetrating oil start flying around. When this stuff gets into your eyes, it has the ability to displace the water on your eye ball. It also can penetrate into pores as small as .00005". Which means the porosity of your eye ball is like the craters on the moon to the stuff. So it lifts the water and gets rid of it, and proceeds to soak into your eyeball, and as you try to wash it away with tears, the oil just displaces the tears. Because your nerves in your eye will be quickly and deeply inundated with the stuff, I can say without a doubt, KANO Kroil is the most painful stuff I've ever gotten into my eyes (one time I had Skydrol mixed with it too, and that was worse... lol...) .
It hurts so bad that you won't care about washing it out, all you will care about is screaming, "Kill me, it hurts", and pounding your head into a wall to knock yourself out, to stop the pain. Your neighbors usually can be found pointing and laughing at that point.

....lol ...... Especially if you have mixed it with 10 other kinds of oils in a shot gun fashion. (OK I am exaggerating a little, but hear me now, or hear me later, the stuff really, really, hurts like no other in your eyeballs, so avoid that! 'nuff said.
The trick is, once it "breaks" free, you have it made. So it is always best to use counter intuitive thinking here. Try and drive the post IN first. If its in 7 inches, and you knock it down to 7 1/4, that's not going to make it worse, rather, you have broken it free of it's bonds, and now you can start going back out with it.
The catch is, if its a crummy material, and soft, it will just mushroom, and bend, and it just doesn't work well. On the bright side, that means the material will be easier to cut, or drill.
A cheap air hammer can do wonders. Usually called a "muffler cutter", et al.. Probably $29 dollars at Fleet Farm, etc...They are good because they hit hard, and repetitively. But you need to find a bit that works here too, a splitter won't work, so finding the right "tool" for the air hammer, will take some thought. A pneumatic rivet gun is the same thing, only much more controllable, and much more expensive.
So, the hammer becomes the usual choice. Bigger is usually better. and you can figure that part out. If it doesn't move with the hammer trying to drive it in, you have little chance at pulling it out. If it does move, (mark it before hitting on it, so you know if it moved), then you can proceed to a slide hammer and you are only hours, or days, away from successful extraction, lol.
If you need to go to the next level, then the sawz-all, or the drilling becomes the next avenue. Remember, if you mash it up all the way down into the seat tube, it usually can't be drilled effectively. So make the call, and cut it off cleanly to be drilled, or follow my lead on the sawz-all technique. I'm not going through this one again, but I will link it here. It was in the Schwinn Sting restoration challenge" thread.....
http://www.vintagebmx.com/community/index....wtopic=27010138A
piloted reamer, or a
piloted core drill is best. If you can obtain, or have on hand, either of these, in the right sizes, then you are already probably like me, and have a machine shop of your own, and already knew all of this stuff. If you really want the right stuff though, start with machining catalogs, and aviation oriented places. Those are your search words.
A regular twist drill bit, is a tricky thing to use here, but it can be done. But even on a mill or a lathe, it's tricky. Be careful. ...Or farm the job out.
As for twisting it. If it won't go in by pounding on it, and is still seized, twisting is a bad thing. The post may not be what comes un-seized first when using torsion to try and break it free.
If you got it to move, then it is not seized, ....then....go for it, twist away.
There is always the one who likes to tie the frame to the car bumper, and use a lag bolt in the foundation of the house.... ....whatever, it's not my frame...