As far as I can tell it DOES affect bosses, and it triggers off anything that deals % weapon damage (the damage dealing components of all monk abilities).

Updated Answer

Having soloed inferno recently with a dual wielding setup on my monk, I can now say as a fact that dual wielding is absolutely possible. But note, this highly depends on having two great weapons to use on your monk.

Here are the main advantages for dual wielding over a shield:

• Additional socket on weapon for HUGE critical hit damage boost (This is the most important factor, I've severly underestimated how much DPS having more crit hit damage can provide with modest crit hit chance (~30%).)
• A ton of more stats
• More Life On Hit Proc (This is the second most important factor)
• 15% faster attack
• 15% more dodge

My primary weapon right now is a ~850 dps 250dex 250vit 500 life on hit axe.

My secondary is a ~750 dps 850 life on hit 97% critical hit chance axe.

The main goal of this setup is to have high dps with huge life on hit and high critical hit damage. I'd prefer my primary to be socketed to have even more critical hit, but those weapons with similar stats to mine are too expensive right now and far beyond my reach.

Now that I have 3x more DPS than the old tank build, even though I'm down quite a bit on defense, its easier since I kill them faster so I don't have to tank as long. Also, the really deadly effects such as arcane turret and ground effects can't be blocked anyway, making blocking less useful than killing the target faster.

Finally, dual wielding axes is major style points, imo looks much nicer than axe/shield ;)

Old Answer

This answer is still applicable since having 2 great weapons is really really expensive compared to 1 and a shield.

From my experience, a shield is extremely necessary in late game inferno and ponies (the 1.03 damage nerf has made shield even better by reducing the spikiness of damage). Lets compare and contrast:

Shield Benefits

1. 1000 - 1500 armor (double that if you use deadly reach)
2. 70-140 resist all (70+ res all and another element of choice for stacking)
3. 30-40% block (high block shield with JL + HoC) for 3k damage (note this is after reduction!)
4. 1 free passive slot (you don't have to take guardian's path so you can devote 1 more passive for defense, such as resolve)

Overall, you gain HUGE damage reduction benefits for 1 equipment slot. If you are trying to be a tank this is almost a no brainer.

Duel Wield Weapon Benefits

1. 15% Faster attack speed for more LoH
2. More stats (weapons can get higher stats than armor can), and more unique effects like transcendence (having 100+ life per spirit spent is a great way to heal).
3. Faster spirit regen (even faster if you have 2 spirit regen weapons)
4. 15% dodge (if you take the passive)

Overall, you gain a little dodge (doesn't work when frozen, etc, and you can't dodge ground effects like desecrate and plague, which are the main sources of death in late game inferno), some more potential sources of healing and spirit regen, more damage, but way less survivability. Either way you won't be a tank.

To summarize, using a shield is a lot more defensive oriented, and thus provides the most survivability in late inferno acts (3+ and ponies). Not to mention finding a good shield is a heck lot cheaper than finding a good 1 hander with high damage, ias, stats, and LoH. You might be able to get away with duel wielding up to act 2, but I wouldn't expect you to last long beyond that unless you have godly gear and weapons.