Ramifications and possibility of a diet consuming exclusively of human semen
No, you cannot live off of semen.

A source from Columbia.edu:


  A typical ejaculation fills up about one teaspoon; the actual amount is determined by a man's age (younger men usually make more semen), when he last ejaculated, and how long he's aroused before ejaculating, among other factors. Contrary to what you've heard, semen is not loaded with calories. Each teaspoon of ejaculate has about 5 - 7 calories and some 200 - 500 million sperm. Since sperm make up only about 1 percent of semen, what accounts for the other 99 percent? Well, its other ingredients include:
  
  
  Fructose sugar
  Water
  Ascorbic acid (a.k.a., vitamin C)
  Citric acid
  Enzymes
  Protein
  Phosphate and bicarbonate buffers (bases)
  Zinc
  
  
  Can swallowing semen enrich a protein poor diet? Unless you're gulping gallons of it each day, it's no substitute for real nutritious cuisine! 


From a non-academic, but apparently well-informed source, semen contains:


  Semen contains only 15 calories and boy does it have a lot of stuff in it. So let's have a look. Ejaculate contains ascorbic acid, blood-group antigens, calcium, chlorine, cholesterol, choline, citric acid, creatine, deoxyribonucleic acid (DNA), fructose, glutathione, hyaluronidase, inositol, lactic acid, magnesium, nitrogen, phosphorous, potassium, purine, pyrimidine, pyruvic acid, sodium, sorbitol, spermidine, spermine, urea, uric acid, vitamin B12, and zinc.


I've highlighted the nutritionally-relevant ingredients. You'll notice that there is some overlap - with the Columbia source. What you won't notice, though, are the rest of the Essential Vitamins and Minerals:

Vitamin A (retinol)
Vitamin B1 (thiamin)
Vitamin B2 (riboflavin, vitamin G)
Vitamin B3 (niacin, vitamin P, vitamin PP)
Vitamin B5 (pantothenic acid)
Vitamin B6 (pyridoxine, pyridoxamine, or pyridoxal)
Vitamin B7 (biotin, vitamin H)
Vitamin B9 (folic acid, folate, vitamin M)
Vitamin D (ergocalciferol, or cholecalciferol)
Vitamin E (tocopherol)
Vitamin K (naphthoquinoids)
Chloride (Cl−)
Chromium (Cr)[4]
Cobalt (Co) (as part of Vitamin B12)
Copper (Cu)
Iodine (I)
Iron (Fe)
Manganese (Mn)
Molybdenum (Mo)
Nickel (Ni)
Selenium (Se)
Sulfur (S) numerous roles[5]


I should also mention that the specific amounts of each ingredient will vary from person to person, as diet can play a major role in flavor resulting from the changed nutritional content (i.e. - higher fructose levels).

As perhaps the most reliable source I could find, there is a Wikipedia article with references concerning the makeup of human seminal fluid and has this information:


  amino acids, citrate, enzymes, flavins, fructose (2–5 mg per mL semen,3 the main energy source of sperm cells, which rely entirely on sugars from the seminal plasma for energy), phosphorylcholine, prostaglandins (involved in suppressing an immune response by the female against the foreign semen), proteins, vitamin C
  
  acid phosphatase, citric acid, fibrinolysin, prostate specific antigen, proteolytic enzymes, zinc 


It echoes the other sources closely, and still yields insufficient evidence that a person could live off of semen exclusively.

As for how long you can live? From a Scientific American article concerning starvation there have been confirmed instances of hungers strikes lasting up to 40 days. Assuming you could consume enough semen to sustain your body's daily Caloric requirements, you would die of malnutrition eventually. Scurvy - the inadequate intake of Vitamin C (Ascorbic Acid) - would probably be the culprit, as deficiency becomes clinical in 1 to 3 months with the later stages of Scurvy being: "...more severe and life threatening; common manifestations include generalized edema, severe jaundice, hemolysis, acute spontaneous bleeding, neuropathy, fever, convulsions, and death."

Assuming that there is enough Vitamin C in semen, you would die of other deficiencies. Since Vitamin B1 (Thiamine) is water-soluble (is not stored in your fatty tissues), not found in semen at all, and the body only reserves about a month's worth, after 30 days you would start to experience Beriberi - which is only slightly less fun than Scurvy.