Adjusting module weight: what are the risks and things to look out for?
Well there isn't a general answer to this, it really depends on the modules. Some of the things I would take into consideration (In Drupal 6) :


hook_boot is called as modules are loaded ; so only modules that are lighter than you have been included. If you rely on code from another module that is heavier than you, make sure you include them manually. 
If a module has by default a weight other than 0, then it means that this module specifically set this weight. Check that module's .install file, as this is most likely where the weight is set - and with a bit of luck there will be a comment explaining why ! (this may be true even if the weight is zero)
Modules that set either a very low or very high weight usually have a good reason for it (must run before/after everything else). Typically development modules do this, though others might as well. Keep these modules first/last unless you have a good reason for it.
Look at the module's dependencies (the ones the module depends on, and the ones that depend on this module). Ideally keep these modules in the same order. If you must change the order, the only way to understand the consequnces is to look at the code.
Use common sense. Information that is added by a module (say to a node on node load) will not be available to modules lighter than this one.