The story of a couple of stuck seatposts is recounted in
this thread. The good news is that they both came out in the end.
Things to try:
- Powerful twisting. If you can fix the seat post in a really stout, immovable vice, and twist the bike, it may get it moving.
- Percussion - with the bike frame resting on something immovable and non-bouncy (so, not its tyres, not a wooden floor), hit the seat post hard. No, much harder than that. You'll be knocking it into the frame, not out of it, but again, once it's moving it may be persuaded to move out as well as in.
- Heat/cold. You want to expand the frame or shrink the seatpost. So hot water on the frame might help, or ice or something cold inside the seatpost. Counterintuitively, heating the seatpost might help too - anything that cracks the 'weld' between it and the frame may get it moving. So a hot air gun on the seatpost might just help, then try again when it's cooled down again.
- Caustic soda. This did the doing for me. It's sold in ironmongers and DIY shops as a powder or pellets, which you make up into a solution (carefully!) for cleaning drains. Take care with it and follow the instructions - it's nasty stuff. If you can figure out a way to introduce it to the interface of frame and seatpost and keep it there, it should dissolve the oxide that binds them together. (Eventually, it would dissolve the seatpost too). My seatpost was fluted, so I just kept squirting a little soda solution down the flutes. You might need to turn the frame upside down and fill the seat tube up with the stuff if yours isn't fluted. You might need to drain and replace the solution periodically - the oxide seems to neutralise the solution after a bit. It stops fizzing anyway.
- Persistence, combining any or all of the above until something gives.
- Bike shops have various techniques and usually better tools than you have. Some of these techniques are fairly brutal, and they'd probably rather you weren't watching. Maybe you'd rather not be there either.
There's an excellent chance that the seatpost will not survive the operation. Be prepared to sacrifice it if necessary. But you'll want to save the frame, and the paint on it, so take care with heat guns and hitting things, and think about what you're doing and what the consequences might be before you do it.
In future, grease the seatpost before fitting it, and remove, clean and re-grease it every year or so, or after a salty winter.