Is there a limit of how close a runway can be to the water?
FAA Advisory Circular (AC) 150/5300-13A covers "standards and recommendations for airport design." It does not include any restrictions on proximity to bodies of water, but it does make the following recommendation in paragraph 319(a):


  It is recommended that the entire RSA and RPZ be accessible to rescue and fire-fighting vehicles such that no part of the RSA or RPZ is more than 330 feet (100 m) from either an all-weather road or a paved operational surface. Where an airport is adjacent to a body of water where access by rescue personnel from airport property is desirable, it is recommended that boat launch ramps with appropriate access roads be provided.


The actual FARs (14 CFR 139, Certification and Operations: Land Airports) also do not state a limit. The only reference to bodies of water of any kind is in §139.325 (Airport emergency plan) paragraph (e):


  The plan required by this section
  shall contain provisions, to the extent
  practicable, for the rescue of aircraft
  accident victims from significant bodies
  of water or marsh lands adjacent to
  the airport which are crossed by the
  approach and departure flight paths of
  air carriers. A body of water or marsh
  land is significant if the area exceeds
  one-quarter square mile and cannot be
  traversed by conventional land rescue
  vehicles. To the extent practicable, the
  plan shall provide for rescue vehicles
  with a combined capacity for handling
  the maximum number of persons that
  can be carried on board the largest air
  carrier aircraft that the airport reasonably
  can be expected to serve.


So, bottom line: there is no limit, but airports are required to take water into due consideration when developing emergency plans, and the FAA looks at each airport and plan on a case-by-case basis during airport certification.

As for the rest of the world, I have no authoritative information; but as others have posted, it certainly seems like the situation is the same, there being numerous examples of airports worldwide built on natural or reclaimed land very close to water.