Can Lasik Cause Eyelids to Sag?

LASIK per se is not the cause of sagging eyelids but having LASIK might be. How can this be so? It is because for LASIK there is a lid clamp that opens the eyes widely and it can stretch the lids in a way to cause ptosis or droopy eyelids although this is a rare complication.
This is one more reason why I like the all laser method of doing LASIK. In particular we are using the newest Zeiss techology called VisuMax which only requires a very slight opening of the eyelids to accomplish the first step of making the flap. The Intralase is also better than the blade method concerning the risk of ptosis, but not as safe as the VisuMax.
The LASIK or reshaping of the eye itself cannot cause any damage to the eyelids and thus no droopy lid. If the eyes are irritated temporarily after lasik or if there is some light sensitivity then there can be some squinting, but this not what you seem to be talking about.

I have never seen this and cannot think of a scenario where this would happen. There are some reports of steroid eyedrops causing a temporary relaxation in the eyelid muscle, which could lead to a slight lid lag. We do use steroid drops post LASIK for a brief time, but, again, I have not seen this as a side effect.

LASIK, and any other type of ocular surgery, CAN cause the eyelids to droop. This is a (rare) complication called ptosis. It is thought to be a possible side effect of the eyelid clamp used to keep the eye open. With cataract surgery (not LASIK), studies have shown a risk to be generally less than one percent. Note that hard and rigid contact lenses can also cause this.
Ptosis is a change in position (a dropping) of the eyelid margin. If you are describing more skin and fat overhanging the lid margin, with no change in the lid margin position itself, you are describing a different condition called dermatochalasis. There is no known association between LASIK and dermatochalasis.
The treatment for dermatochalasis is blepharoplasty, commonly known as an "eye lift." This is very different from ptosis repair, which will alter the position of the lid margin itself.