How to remove solder from via?

Have you tried a desoldering pump?

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Beyond what has already been answered, you can try to use wick solder remove.

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There is a great article at wikipedia:

And Youtube has many videos about:

Sometimes you can just push the majority of the stuff out, by quickly pushing a couple inches of un-heated solid wire through the hole while the solder is melted. If you do it fast enough, you end up with the solder frozen on the wire sticking out the other side of the board, and you can break the solder off and then just pull the wire back through.

Another thing that sometimes works is to heat the via and then quickly tap the board against the edge of a bench, moving the board in a short swing in the direction of the via. The inertia of the molten solder will cause it to keep going forward (and probably onto the floor) when the rest of the board stops. And (obviously, I had thought) you don't want use this technique on any but the sturdiest of boards.

Like the blowing technique, the down-side of these is that they can produce tiny, loose bits of conductive material to watch out for, which can cause shorts if you don't get them all. Suction is really the way to go.