Lone pairs can sometimes be delocalized. An acceptable rule of thumb is that if an atom possesses lone pairs and is connected to an sp2 atom, then the two atoms participate in resonance. So consider phenol. Oxygen has lone pairs. It's connected to an sp2 carbon. And so yes, the oxygen can participate in resonance with the sp2 carbon it's connected to, and also the other sp2 carbons in the aromatic ring. We can ascertain the fact that every atom is phenol is sp2 by noting that phenol is planar.
In general, a negative formal charge makes delocalization a (more) favorable move as this decreases the electron (charge) density on one atom. Spreading out electron density between or among multiple atoms generally lessens reactivity.
Now, your question is why would oxygen exhibit a +I or electron-donating effect despite its high electronegativity. Well, it can participate in resonance donation as explained above; see phenol and its conjugate base, the phenoxide ion, for examples. Examining both in terms of electrophilic aromatic substitution also reveals that the oxygen atom in both is strongly electron donating. True, oxygen can withdraw electron density via sigma bonds (inductive effect) but doing so is unfavorable especially in the phenoxide ion in which the oxygen already bears a negative formal charge. As your teacher explained, bearing charge isn't something that is very favorable.
And in general, you're going to find that resonance effects outweigh inductive effects. Hence the reason why an sp2 oxygen connected to another sp2 atom generally exhibits a +I effect. Consider dimethyl malonate and methyl acetoacetate . Which is more acidic? From inductive effects alone we'd say dimethyl malonate because the oxygen is highly electronegative as opposed to an sp3 carbon and should withdraw more electron density via the inductive effect. From a resonance-standpoint we'd say the opposite; the oxygen in the ester donates electron density via resonance more than a methyl group can inductively donate electron density. And you'd have to realize that the resonance-standpoint is more important in evaluating the acidities of the two weak acids.
A common exception to resonance>inductive effect in organic chemistry is an sp2 atom connected to a halogen. Yes, the halogen has lone pairs and these can be delocalized into the adjacent p-orbital. But the halogens (except possibly fluorine) are much bigger than many other atoms, and therefore the magnitude of resonance donation is limited. So the great electronegativity of the halogens trumps their magnitude resonance donation, and we can see this in the context of electrophilic aromatic substitution again; halogens are ortho-para directors (delocalize the lone pairs in the Lewis structures to see why) but are also mild deactivators.