TesterMen Tester provided some very good information, which I think requires a bit of explanation. The older version of the regulation seems to require fuel dump if MTOW is more than 105 percent of MLW. None of the 737 versions have weight limits meeting that requirement, so based on that regulation, it seems like fuel dump would be required. But as the Boeing info shows, none of the 737 versions have fuel dump capability.
That regulation was amended in 1968 to allow a plane to not have a fuel dump system as long as it can meet certain performance requirements. The current version of the regulation includes this as well.
However, this does not explain why the 737-100/200, which were certified before that amendment, don't have fuel dump capability, when the MTOW listed by the Boeing info above are 108% and 112% of the MLW.
The explanation is that the original versions of the aircraft had lower MTOW, which met the 105% rule. Newer engines allowed the MTOW to be increased, but as rbp mentioned, MLW is based on landing gear and structural limits, so it did not change. The FAA amended the regulation with the required performance option to allow the higher MTOW without requiring fuel dump capability. Also, as explained in answers to this related question, planes can land over MLW, there just may be required inspections.
Note that the similar-sized A320 family does not have fuel dump capability either.