The reason I ask is since my buddies and I watched Saving Private Ryan yesterday. I know it's not a great idea to draw historical inferences/make historical criticisms based on films, but it really seems like the a lot of lives wouldn't've gone to waste had the Allies used naval artillery or air-dropped munitions to deliver some sort of smoke screen onto the beaches in order to obfuscate views from German firing positions. I know I'm being an armchair general here, but was there a reason for not employing this tactic? Do you guys think it could've saved lives?
Screening smoke using the standard smoke composition lasts a relatively short amount of time. The volume is also not infinite.
Attacks by Naval fire or aircraft would have to be extremely precise and well timed to put smoke in the proper lines of sight between Axis positions and allied landing craft. Perhaps impossibly so, given both their technology and military intelligence level the allies had of axis positions. For the amount of effort and time spent doing this, it is much more advantageous to simply drop explosives on the enemy if you have the ability to be so precise.
Mislaid smoke could also serve to hinder allied advance just as much as aid them. Smoke on the beach could potentially stop the attackers from being able to fire out at the dispersed, hardened positions of the defenders, while the defenders would have much more success blindly firing into the smoke from their static positions. Smoke causes disruption to rapid movement and organization if laid wrong. The allies needed to be able to move and communicate more than the defenders on the beachhead needed it.
On D-Day the allied aircraft and naval artillery were working to massively bombard the Axis defensive positions with explosives in an effort to kill and permanently disrupt their defense, which was more important than attempting to smoke positions for a temporary gain.
The link that /u/losphobos provided shows why the smoke screen would not have worked well and if did it would have caused bigger problems.
You'll note that the smoke in that picture is being blown strongly to the left (East) of the picture. As others have pointed out artificial smoke only lasts a very short period of time. If you've read about the D-Day landings at all you'll know that a storm front had passed through the day before. The Westerly winds from that system were still blowing hard on the morning of the 6th and would have ripped any artificial smoke screen to shreds.
Incidentally most of the smoke in that picture is from fires which were started on the bluffs by rockets launched from modified landing craft just minutes ahead of the assault waves. The smoke did provide some cover for troops landing downwind.
Which highlights the other reason for not using smoke:
The landing craft crews needed to be able to see the various terrain features like gullies and church steeples in order to steer their boats to the correct landing point. The winds and incoming tide pushed most of the landing craft to the East. Unable to see their landmarks clearly the crews could not adjust appropriately for the drift. As a result very few units landed anywhere close to the their intended landing point.
Joe Balkoski, Omaha Beach
https://en.wikipedia.org/wiki/Omaha_Beach#/media/File:Into_the_Jaws_of_Death_23-0455M_edit.jpg
So if you look at this famous image from the invasion, you can see that there was plenty of smoke generated from the naval bombardment, though I can't say weather or not this involved smoke rounds specifically.
The weather was such on D-Day that Aerial Bombardment was ineffective, as cloud cover prevented accurate bombing of shore defenses. There was a preliminary naval bombardment the preceded the assault by 40 minutes or so, but you have to keep in mind that smoke doesn't linger for long, and that there is a risk of injuring your own soldiers if you're shelling the beaches at the same time as the landings. (I read on wikipedia that an effective smoke haze was maintained at the harbor of Anzio during it's invasion, but this was probably done using smoke generators once the beach head had been established, to hinder enemy artillery.) It's also worth keeping in mind that smoke would obscure the vision of attacking troops and vehicles, so it really seems like a double edged sword.
(Disclaimer: I'm not a historian, only a bit of formation in chemistry)
Smoke screens work by creating a dense aerosol. To create such an aerosol, you either disperse an agent (like it is done in smoke generators and phosporous-based smoke screens), which will be either not very dense or not very long-lasting, or you disperse a highly hygroscopic agent, which will basically "grab" the ambient moisture in the air and create a dense fog.
Ruling out phosphorous due to it's low endurance and hazard to your own troops, we could look for the second kind. Maybe you have come into (hopefully not direct) contact with disulfuric acid - you will have noticed that it appears to fume. That's basically what happens with the second kind of smoke agents. The trouble is that these fumes are highly acidic and - depending on the kind of reagent - might contain other nasties such as phosgene. You do not want to drop that on your troops and expect them to fight. The density might be okay, endurance too, but the side effects would more than probably outweigh the benfits.