Do emptied loot chests at Dragon Lairs respawn new content over time?

Nope, I have checked several times.

If you go to where a dragon is, and kill the dragon, get the word wall, and empty the chest, you are done there. Later on, I do not know how long, but you happen to go back to that same spot, a dragon will have re-spawned, and you can kill it, but there will be no dragon soul. The dragon will not burst into flames, but the dragon will have plunder on it. The word wall will not give another word, but the chest will often re-spawn with plunder.

A dragon priest lair will not re-spawn the plunder in a chest, but I have had the dungeon itself re-spawn. There are enemies and depending on ones level, they will be harder and there could be more deathlords than before. I do not remember if the dragon priest itself re-spawns, but I do know the dungeon does.

I think it's 30 in game days before the chests respawn if I remember correctly.

Your slightly off, you can still absorb the dragon soul No matter how many times it respawns, and it still turns into a skeleton. Also the Scripted Dragon Priests like Krosis never ever respawn again. Depending on your level you can find regular unmasked dragon priests in select locations though.

I have been to the locations several times and the chests are always empty, and Well over 30 days have past considering I still play with my original character.

Normal chests respawn in 30 days, but the dragon treasure chests don't. I don't believe the main treaure chest in most bandit and draugr dungeons respawn either. The bandits and draugr usually do respawn though, and I've killed secondary dragons at walls and always gotten a soul. I believe that named dragons don't respawn though...

The chests do respawn after some time, I don't know how long though.

But something I discovered was that many different things would not respawn in a play through where I completely refrained from using any kind of fast travel.

Here's an experiment I made: I loaded an old savegame and went to one of the Dragon Lairs I had cleared. I checked the chest, and it only had two items in it (a pair of gauntlets that I probably left cause I didn't want them). When I visited the same place in one of my recent saves, the chest had new contents (including gold).

30 in game days.

I checked it.

Chests seem to respawn after hitting another level stage
(1-10 and 37-47 for example). Respawning will then take a few days. Including the the dragon, which will be a higher level dragon(from normal dragon to blood dragon for example). The dragon will give another soul. Scripted dragon priests never respawn. I'm not sure about the 30 days respawn for the chests.

I tried it on PS3 vanilla Skyrim. No DLC.

Correct, 30 days for a cleared place, 10 days for a non-cleared place.

Nope, it's just related to time passed.

I have my dragon chests respawning, but then again, I use official patches, so maybe they changed that. :s