Faery Fire: "Outlined subjects shed light as candles..."
It penetrates Darkness, but it is still a sight-based effect.
Obscuring Mist: "The vapour obscures all sight, including darkvision, beyond 5 feet."
I would suggest that Obscuring Mist prevents line of sight, though not line of effect beyond 5 feet; you cannot see far enough to see the (admittedly magical) 'shed light as candles'.
In terms of RAW Faery Fire describes changing the appearance of the targets, and Obscuring Fog 'obscures all sight' beyond 5 feet; you cannot see far enough to see the changed appearance. Faery Fire does not specifically mention countering Obscuring Mist; Obscuring Mist does not have the Darkness subtype, and other than darkness all of the examples given as being countered by Faery Fire effect individuals and are illusions, were as Obscuring Fog effects the terrain and is a conjuration.
In terms of the real world (not necessarily relevant) you might see a step up of 5 feet to their sight (i.e. 5 feet = normal vision, 10 feet = concealment, 15 feet = total concealment), from trying to see a candle in a fog, but not much more than that.
Mechanically, you can't see in Darkness because there is no light to see by - Faery Fire provides that light*. Similarly invisibility and the other similar effects are countered by the Faery Fire's halo of light surrounding the targets. But Obscuring Mist puts hundreds water droplets in between you and what you are trying to see, obscuring your vision; effectively there is an object between you and what you are trying to look at - much like I wouldn't let Faery Fire let people see through natural rock, I wouldn't let Faery Fire let people see through natural fog.
I would say that the suggestion in the paragraph two above (5 foot step up) would be a not unreasonable, but still generous interpretation, and that no change (from Obscuring Mist) would be a reasonable and, in my opinion, RAW interpretation.
I would suggest that blindness would similarly prevent seeing a creature with Faery Fire on it. As with Obscuring Mist, mechanically, Faery Fire does not counteract the effect preventing you from seeing the target - in both cases you cannot see the Faery Fire for things to be highlighted by it.
For a target with both Blink and Faery Fire I would suggest it was as if you could see an Invisible target with the effect Blink (20% miss chance), because Faery Fire provides mechanics to see creatures with invisibility, but not plane shifting.
*Deeper Darkness has an additional effect of suppressing lower level light effects, like 'light' and Faery Fire. From Faery Fire: "(though a 2nd-level or higher magical darkness effect functions normally)" - Obscuring Mist is not a Darkness effect.