What complications can come from a blepharoplasty? How do you handle them?

Generally, upper eyelid Blepharoplasty is a very high satisfaction, low complication type procedure. Lower lid Blepharoplasty has a greater complication rate, but should still be very low in experienced hands.
However, given the question, I will give you the standard run down of complications I explain to may patients (informed consent). Realize that informed consent is a process which allows you to understand exactly what you are proceeding with, not to scare you, but to make sure you are fully educated.
Upper Eyelid:
1) Over resection of skin and inability to close the eyes. I explain to patients that this could happen, but is very easy to avoid. I pinch the eyelid skin while the patient is awake to determine the maximum amount of skin I can safely remove. I then take a little bit less than that.
2) Dry eyes. As we age, we produce less and less tears; every patient gets a temporary dry eye syndrome after eyelid surgery and artificial tears are required. A permanent, clinically significant dry eye syndrome is possible albeit rare and probably more related in nature to over-resection of skin.
3) Asymmetry of eyelid creases. Generally speaking, the lower part of the upper lid incision is made along the upper eyelid crease and correct marking of skin and incision placement avoids this problem. This problem is more likely to occur if the surgeon wishes to reset the upper eyelid crease.
4) Unmasking of pre-existing ptosis. If ptosis (droopy eyelid) is not recognized preoperatively, it generally becomes fairly obvious after the surgery.
5) Bleeding, infection ,scarring. Bleeding is minimal. I have never seen an upper lid incision become infected. Upper eyelid skin is the thinnest in the body and typically forms excellent scars.
Lower eyelid Surgery:
I mention most of the complications above first.
1) Lower lid retraction and/or ectropion. This applies primarily to external approaches, not to transconjunctival approaches. However, in my opinion, an external approach is frequently required to address the tissue descent and skin excess seen with the facial aging process. If excess skin is removed without adequate muscle suspension, the lower lid will lose the battle of the scarring process with the cheek. In severe cases, this can lead to ectropion and corneal exposure. This is an avoidable complication based on experience and judicious technique. It is also more likely to occur in revision cases.
2) Loss of Eyelashes. This is possible but very avoidable via judicious tissue handling and incision placement.
3) Loss of vision. This has been described in the literature via case reports; these reports mainly come from the time period where the anti-platelet activity of aspirin was not recognized, and in cases performed where uncontrolled hypertension was not addressed preoperatively. I have never seen this happen in my community but I always mention it so that a patient is fully informed.

Great question.  Every patient should be well aware of all possible expected post op recovery outcomes as well as possible complications.  Some normal expected post operative issues after blepharoplasty include swelling and bruising that may last up to 2 weeks.  This may resolve more quickly with icing for the first 48 hours, taking oral and topical arnica, and elevating the head when sleeping for the first week or so.   It is normal to have temporary dry eye symptoms (either the eyes feel dry and ichy, or they may seem watery (watery eyes may be a symptom of dryness).  Using plenty of antibiotic ophthalmic ointment at night and artificial tears during the day will help with this.  Chemosis is when the whites of the eyes become swollen. This may be a post op side effect as well - if you develop chemises, your doctor may prescribe a steroid eye drop. 
Complications may include infection, bleeding in to the orbit (eye socket), worst of all and very rarely blindness.  If after surgery your pain gets worse rather than better, or if you notice increasing pressure, pain in one eye and blurring or loss of vision, these are emergency concerns for which you should call your surgeon immediately and be evaluated in the emergency room.  Blood or clot trapped in the eye socket can increase pressure on the globe resulting in vision loss if the pressure is not released urgently.  

Every medical treatment has risks, and this is true for eyelid surgery as well. Risks of eyelid surgery include bleeding, scarring, dry eye, infection, difficulty closing the eye, corneal abrasion, inadequate (or overzealous) surgical excision of either skin or fat resulting in patient dissatisfaction, eyelid malposition, need for future surgery, anesthesia complications, and more. As long as you choose a well-qualified and well-trained surgeon who can demonstrate experience in the procedures, these risks are generally very low and great results with eyelid surgery are possible.
As with any surgical procedure, if complications do occur, the first step is to notify your surgeon of any concerns and follow their recommendations very closely.
Be sure to see an expert who specializes in facial aesthetic surgical procedures.
All the best,

A blepharoplasty is a wonderful procedure with an amazing outcome and quite low complication. I always perform my signature TBT-eyelift procedure which is a less-invasive and less-traumatic procedure which minimises the risks and complications. Instead of removing the orbital fat which needs a resection/opening of the orbital septum I reposition the orbital fat and leave the septum intact. This septum divides the eyelid from the orbit with its very delicate structures. With the traditional blepharoplasty a major risk is the over-resection of orbital fat which creates a hollow-eye look of the upper eyelid which looks either very operated or even very old (since we loose this fat with ageing). When resecting the orbital fat the lacrimal glands which look similar to the orbital fat could be hurt or removed resulting in a dry eye for life. Bleeding fat might glide back into the orbit after a part was resected and could cause pressure on the optical nerve which can cause blindness. These risks are not involved with the TBT-lift since the septum is not destroyed but only tightened with an external suture. Further complications of an upper eyelift could be a temporary dryness or chemosis (irritation) of the eyes. Risks like conjunctivitis are very rare and more likely occur when an upper and lower eyelift are combined. An assymetric result is very rare but can easily corrected in a minor revision surgery.

These vary from the upper to the lower lid.  The most common is more than expected bruising and swelling.  Chemosis or swelling of the skin of the eye can occur, as can dry eyes. eyelid closure issues, and irritation.  Worse complications are changes in the shape of the eye, pain, chronic redness and tearing.  These are less common unless heling is delayed (rarely occurs).  These are the basics but the discussion requires a lengthy review.  You can go over this with your surgeon prior to surgery.

I am not going to go through a list of technical complications of the surgery, as many of my colleagues have already done so at length. As a blepharoplasty surgeon specialising in revisions and redos, I would like to offer a different perspective.
Although complications like eyelid retraction etc do occur, by far the most common complications are related to the eye surface, particularly dye eye and blepharitis. Blepharoplasty is unique among cosmetic procedures in that most of the surgeons who offer this procedure have no training in ophthalmology. This means that if there is an underlying problem or complication affecting the eye, they have no knowledge of what might be going on, no training in examining the eye and no concept of how to manage or when to refer. 
As an oculoplastic specialist, I have been trained in ophthalmology first, meaning that I have a deep understanding of eye disease. An essential part of the preoperative work up is the actual eye examination which is performed with biomivroscopy and is essential to anticipate potential eye problems. Only an ocloplastic specialist can do this. 
There is no doubt that plastic surgeons have great tissue handling skills etc and can execute the surgery well, but given at they have no undstanding or ability to assess and manage eye problems, I would strongly advise using an oculppalstic surgeon.
To summarise this message, I would say that the complications not really meantioned by others are the most common, which are ocular surface disorders. Theses can lead to chronic discomfort, pain and sometimes visual problems. I have dealt with a huge number of these cases that have been left untreated and could have been anticipated and avoided if carefully examined and prepared before surgery. If you want the confidence of knowing that your eyes are also being taken care of, use an oculoplastic specialist.