(This question pertains, essentially, only to snakes, which are the main
consumers of feeder rodents.  Although some lizards and amphibians will eat
rodents, amphibians typically will not take dead food, and most carnivorous
lizards eat rodents too small for the concerns of this section to be a
factor.  Large monitors are an exception, and this question may apply to
them as well.)

Although a snake is a pretty formidable adversary for even the toughest
rodent, a feeder can occasionally get lucky and manage to bite its predator.
Such bites can be serious; in extreme cases, the rodent can land one
fortunate bite at the base of the skull and kill the snake outright.
Most feeding bites are much less serious and pose no real threat except from
infection, but such catastrophes really have occurred.  This is one very
good reason to prefer to use dead feeders; a prekilled mouse will rarely bite
a snake.  This goes double for gerbils, which are fast and scrappy, and at
least triple for adult rats.

Another convenient feature of prekilled rodents is their availability; it
is possible to mail-order hundreds of frozen rodents, fill a freezer with
them, and have a practically permanent food supply for your snakes.  Many
of the rec.pets.herp regulars (the author included) do precisely this.  It's
convenient, and also much cheaper than buying individual live rodents at
pet-store prices.

Most snakes of commonly-kept species can be conditioned to accept prekilled
prey, though the conditioning process is sometimes lengthy and frustrating.
The tricks used to encourage feeding are innnumerable and really beyond the
scope of this FAQ, but often simply wiggling a dead feeder (with a pair of
forceps---don't use your bare hand or you *will* get bitten) is enough to
interest a reluctant snake.

Some snakes simply refuse to eat anything other than live prey.  It behooves
the responsible herp keeper, when faced with such a specimen, to take every
precaution to make sure the predator-prey relationship doesn't reverse itself
(and, yes, there *are* cases in which snake keepers have found an intended
feeder rodent making a meal of the snake)!  Never leave a live feeder rodent
alone with a snake, especially in the case of tough scrappers like rats.  If
possible, stun the feeder before offering it; many snakes that turn up their
rostral scales at prekilled prey will still eat live but unconscious animals.
In short, don't invite trouble.

Naturally, many of the caveats of this section do not apply to pinky or fuzzy
rodents, which are not yet developed enough to injure anything larger than a
small insect.  However, conditioning a snake to take prekilled pinkies or
fuzzies while it is a juvenile may help encourage it to eat dead prey as an
adult.

In the first draft of this answer, I wrote "A prekilled mouse will never bite
a snake."  I'm wrong; in March 1996, a poster actually reported seeing his
corn snake receive a "bite" from a dead mouse!  The snake managed to knock
the mouse's mouth open and drag the teeth over its side while searching for
the head.  (Fortunately, the injury was extremely minor.)  This anecdote
should only strengthen your resolve to feed prekilled; if even a *dead* prey
item presents a slight hazard, just imagine what a *live* one could do!

Legislation affects the use of feeder animals in the UK (the Protection of
Animals Act) and perhaps other countries as well.  The UK law is not
particularly restrictive---it requires that live feeder vertebrates be used
only as a last resort and that the feeding process be monitored.  Local US
jurisdictions may also have relevant regulations.  Apprise yourself of the
local legislative situation as it applies to your feeding practices.