This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
It all depends on the surgery. For example - could I suspend you in
transport, reform the pattern so that your arm is no longer broken,
your skin is no longer cut, etc? Yes. But in sickbay they already have
machines which do it near-instantly, and don't take the massive
resources of the transporter.
For such things as removing a tumor, you must consider what replaces
the object being transported away. In all likelihood, a vacuum. Having
a small vacuum appear inside you body is probably more deadly than the
tumor was in the first place. It has been suggested that you could
synchronize two ACBs and beam in a saline solution in place of the
tumor you are transporting out, but again, why bother? There are
already medical devices which probably use micro-transporter
technology to effect the surgery.
In "Deadlock" [VOY], a baby (Naomi Wildman) was transported out of its
mother when there were complications during delivery - so apparently
this sort of thing is performed in emergency situations when the
normal medical resources are inadequate for the task. (Thanks to Eur
van Andel for pointing that out.)
 
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