How important is IV, or, what IV is sufficient?

Gym attacking - it doesn't really matter, as a few power ups extra will make a low IV attacker just as strong as a high IV attacker. That being said, those extra power ups cost stardust and candy, so it makes more sense to power up higher IV attackers, and the higher the better to save on resources.

Prestiging - IV is irrelevant, as you are just shooting for 1000 prestige against everything but Blissey

Defending - IV is everything if you are maxing a high CP defender and need to get as far from the bottom of the gym as possible. That being said, an 82% Tyranitar is still going to out max CP a 100% Snorlax, so the need varies by species and your areas typical defenders.

Even 0/0/0 IVs wouldn't be observably worse. The only thing you might notice is if you were counting the number of hits before your Pokemon fainted and you were dodging everything, where HP would make a difference.

Attack and defence IVs usually only make a difference to charge moves and a minor one at that. +1 damage to a quick move is the most beneficial but even then it's only against Blissey where it realistically makes a difference since it's the only Pokemon there's a risk of timing out against and where you attack it enough times for those +1s to make a difference.

But only looking at the HP bar go down, you won't notice the difference between your HP bar going 11% down rather than 10% due to your max HP being lower.

If you're looking for a defender it's fine to be anal about IVs. But if you're looking for an attacker where 100 CP is only a difference in text and not gameplay, usually evolving higher level Pokemon to save yourself candy on power ups is the smart choice.

John,
Everything could change if/when Niantic changes the gym play.
but for now here is how I play.

Since CP is currently king in gyms, I try to make sure that the mons that I want to place on top of gyms have the highest CP possible, thus I need the IVs to be as close to 100 as I can get.

Moveset is king when attacking gyms, IVs are merely a blip on the radar compared to the tsunami of importance for the moveset.
As someone posted awhile ago
species>moveset>CP>IV

Gym Defending -- Having a high CP is critical for staying in gyms. CP depends first on species and next on IV. The CP formula weights attack heavily, so attack IV is generally more important, particularly for mons with low base attack, like Blissey. For example, if you need a 3000 CP for a good gym position, then a caught in wild ~60% IV Tyranitar is sufficient on level 30; but a 100% Gyarados would require a much larger stardust investment for gym usage, as it would need to be powered up to lev 34.

Gym Attacking -- In general IV is less important than with defenders, but damage rounding thresholds can be important. For example, in a Tyranitar vs Blissey battle, a 14 attack IV Tyranitar would take ~15 more seconds to win than a 15 attack IV Tyranitar, if the Blissey was ~2.5 levels higher (and had perfect defense IV). This large difference relates to crossing a damage threshold, where bite does 4 damage with 14 attack and 5 damage with 15 attack IV. If you aren't crossing a damage threshold, then IVs are less important, but higher is better.

Prestigers -- In general, IVs are less important than the other categories. In theory, a presitiger with 0 attack combined with perfect defense/stamina is best due to the attack heavy CP formula. However, this can change when crossing damage rounding thresholds, like with attackers.

Important to gym placement and most importantly, trainer's vanity.

For the longest time I used to only keep wonders. Now that I'm past 30 and occasionally catch a high leveled mon I sometimes make exceptions. I caught a wild of 30 machop and Larvitar surfing this event. Both are 65% I will hold on to them.

If they were level 20 - to the prof.!

I would not drop dust on anything below 89% and even then only rare mons with the best moveset. All of my dusted mons are over 90% with the exception being my only Stone Edge Rhyhorn at 89.

I have a few 70-80% rare mons for prestige or attack: Electabuz (dual thunder) Lapras(dual ice) lv30 Aerodactly and Pinsir too.

To add to the great answers here:
Consider creating a simple spreadsheet, using "real life" examples of common mons. I did this for grass attackers prestiging against vapes, and concluded that more often then not there was NO difference in quick move attack damage over a wide range of either levels or IV values - the floor() function smooths these out. Clearly there are exceptions - the T-rex vs Blissey example someone posted - if a +1 threshold is crossed and it's a long battle. The most reliable stat that helps prestigers is stamina, where a 0 to 15 difference is about 10-11 HP (depending on level) - could be crucial to survive additional dodges.
Also, Reddit threads on prestiging assume a "perfect prestiger" at 0 attack / 0 defense/ 15 stamina, to most efficiently max CP differential and thereby gain most gym prestige.
Of course, who knows what will change with the gym rework!

For an attacker I'd say it need not matter. I caught a wild 40 IV DB/DC Dragonite (at 700 CP or so) september last year. Now it is a 2916CP level 31. I have treasured it ever since and it has been through, probably every single gym battle I've attempted after I got it to a useful CP.

Of course, there were those times when that low IV hurt a bit. For example when I could have taken one more quick move before fainting (and that one more quick move meant I could deal another 3 dragon breaths to finish the enemy etc). But it really goes down to marginal situations like that.

For the big picture, no it doesn't matter. As far as an attacker goes, species, moveset matters more than IV.