There is a warlock in our party whose strength was just reduced to 0 by a sea hag. He is underwater.
Can a helpless Warlock activate Fell Flight to move underwater?
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When ability damage reduces a creature's Strength to 0, the creature is helpless, and a helpless creature, in addition to other effects, "is treated as having a Dexterity of 0," which means the creature is also paralyzed. (In short, by treating the creature as if it had a Dexterity score of 0, the creature suffers all the effects of having a Dexterity score of 0 without actually having a Dexterity score of 0; this means the creature can, for example, still suffer Dexterity ability damage. Yay?) So while the creature is not impaired because the creature is "held, bound, sleeping, [or] unconscious," the creature is impaired because the creature is paralyzed and, of course, one suspects, because the creature is "otherwise completely at an opponent’s mercy," and, still further,—due to the creature's Strength score of 0—, because the creature's fallen down and can't move at all.
In short, assuming for some reason the warlock hadn't employed previously the invocation fell flight or assuming that the invocation fell flight had been somehow dispelled, the warlock can't take a standard action to employ the invocation fell flight to gain for 24 hours a fly speed because "all invocations… have a somatic component," and the warlock is paralyzed and can't move at all (and that probably includes both the warlock's hands).
(Weirdly, a creature that has instead its Dexterity reduced to 0 is paralyzed then, because it's paralyzed, also helpless, making the net effect of reducing either Strength or Dexterity to 0 practically identical.)
However, had the warlock already employed the invocation fell flight and—even as the warlock's Strength score is reduced 0—the invocation continues to operate (as it typically should unless dispelled), there are other factors to consider.
A creature that has had its Strength score reduced to 0 can take purely mental actions, but it's unlikely that the DM will rule that flying—no matter how its achieved—is a purely mental act: flying usually entails movement, and the creature is unable to move because of both the condition helpless and the condition paralyzed! Were the DM to rule that a warlock can take a purely mental action to fly, the campaign would have to deal with the fallout from such a ruling: any wingless creature that the DM rules flies by mental effort alone (Tip: That'd probably be all the DM's applicable monsters) is able to fly away despite being affected by a hold person effect, for example. (If it's still unclear, this DM recommends against such a ruling: seriously, wizards are already powerful enough!)
With the warlock's fell flight invocation granting a maneuverability of good (DMG 20), a helpless (therefore paralyzed) warlock should hover in place. However, it's possible for the DM to rule that because the warlock invocation fell flight grants a fly speed via a "winglike cape of shadows" that the warlock flies via this ersatz wing, and a "winged creature flying in the air at the time that it becomes paralyzed cannot flap its wings and falls," although to this player that seems overstepping.
Despite all this, I should point out that a warlock's flight is useless underwater as a movement mode. The Dragon #301 Wizards' Workshop column "Sage Advice: Questions and Answers" includes this exchange:
Can a creature fly underwater? I’ve never heard of a normal bird flying underwater. What are the rules regarding this? If it’s possible, what is the penalty movement and maneuverability?
Flying creatures must swim when they enter the water, just as other creatures do. A fly spell lets you swim at your normal flying speed as other swimmers (perfect maneuverability). Incorporeal creatures also can move in this manner (they move through water just as easily as they move through other substances). Note that gaseous creatures cannot enter water. (108)
This ruling predates the Dungeons and Dragons 3.5 revision, but, so far as this answer's author is aware, this information was never contradicted nor repeated. This means that while the warlock could use the fell flight invocation's speed on which to base his swim speed, the warlock must still make Swim skill checks normally to advance in a liquid medium.
[Edit: This answers the Fell Flight part of your question, not the helpless part]
I'll give this a conditional: No on the fell flight.
Flight is movement through air and burrowing is movement through earth and swim is movement through water. Those are different elements.
I give that answer with a condition because it makes sense to be able to to do it as water is simply a denser fluid than air. So you should be able to fly and very low speeds but it is completely up to the GM since magic might care more about elements than physics.