Typically that will be the case (being an ion is more polar), but polarity of a molecule has nothing to do with solvent - although the polarity may change how it interacts with the solvent.
In your example, the O- would be more polar than the O-H (assuming the molecule is the same). Its nice to imagine the electro-static potenial maps some chemistry books have to decide whether one is more/less polar. So the O- will have much more negative charge than the O-H and will make the molecule overall more polar.
(I dont know for sure, but I think a case where the deprotonated version would be less polar might be benzene with an alcohol substituent which has the negative charge resonance donated.)
With your other example, NR4+, the individual bonds may have some polarity associated with them, but the overall molecule would be classified as non-polar since the Dipoles cancel out int he tetrahedral structure (assuming the R groups are the same; if not, there will be an overall dipole).
To make a statement as to whether hydrogen bonding or charge is more important in determing solubility in a substance is difficult. I would say that generally though, if it is a question of an alcohol vs the deprotenated version, the ion will be more soluble. Comparing dis-similar substances would require you to take the whole structures into account though.
Hopefully I got at what you were asking and this helps.
