How can a polar molecule pass through polar channels of proteins in the cell membrane?
The molecule won't "stick" to the polar amino-acids for at least three reasons : 


The amino-acids might not all be on the same plan, so their is always a force pulling the molecules transiting trough the channel in one direction (in or out the cell)
When  a flux is established, the repulsion forces between the molecules will force them to go in a direction, meaning that a coming molecule will basically push the ones already engaged in the channel.
For an ion or a protein, the water molecules constituting the hydratation shell of the molecule can't always follow it into the channel, so the water molecules present at the end of the channel (entry and end being relative to the direction of the flux) will more or less pull it outside the channel, as it is thermodynamically more stable to have a shell.  The fact that water molecules are able or not to follow the molecule depends on the size of the channel.


It's worth noting that water is itself a polar molecule, and that it can selectively cross the membrane thanks to channels called aquaporins.