Say there is an influential and popular research professor who is moving up fast in his research areas - and that he heads a big research group at his school. Then he gets an offer that he can't refuse - from a big time university. He leaves six months or so later, bringing some of his doctoral students with him, too. In general, has this professor burned his bridges with his former university? I imagine he is bringing a lot of talented grad students and grant money in to the new school - and a lot of grant money is leaving the old school.
In general, no, it is not a problem to leave to a new position after some month. The colleagues know how the game goes and could be in the same situation.
However, in practice it depends very much on the special situation. Some examples where leaving after a short period of time are:
Being hired on the promise to build a workgroup in a new field that is not present at the department.
Being hired to teach a special much needed class.
You promised to submit a large grant proposal that would boost the universities standing if successful (e. g. NSF centers, DFG Sonderforschungsbereich or such).
You got a large startup funding, spend a lot of money on specific equipment, that is of no use to anybody else.
You started to supervise many students and other colleagues have to step in to help out.
Probably there are more things, but the bottom line is "Don't be a jerk and you'll be fine".
In principle no but in practice the decision to leave may lead to some tensions, even if @Karl and @Dan Romick rightly point out that it should not be so.
There is always a element of risk involved in hiring someone: a department has usually invested some time and committee work to hire this person, possibly had to lobby the administration for the position, may have supported this faculty with teaching buyouts so she/he could establish a research program. When this investment suddently disappears, very few will normally congratulate themselves at the thought of starting the process again (especially if the competition to hire the faculty now leaving was hot, and there were multiple good candidates).
Many no doubt will be sad to see a friend leave: there is no more reason to burn bridges than if a good neighbor moved to a new town.
Of course there are people who are truly hated (for legitimate or jealous reasons) and everyone's happy to see them go. In the same category some researchers feel they are not getting the support they rightly (in their minds) deserve. In these rarer cases, the departure can be acrimonious.