Are there experiments that are banned from being done at the LHC because they are too dangerous?
Concerning mini black holes for example, that would evaporate in the course of a Planck time anyway, you would need a linear accelerator of the size of a galaxy to produce them as Lenny Susskind explains for example.
Any claims that the experiments done at the LHC are dangerous are blatant nonsense spread by people who just wanted to get their minute of fame, media that misunderstood what serious physicists said or dishonestly distorted it in order to better sell their stories etc, ... Nobody should listen to such sensationalist screams of people who have no idea what they are talking about.
Nevertheless, there is an article about the safety of the LHC wherein the (non existing) danger of mini black holes, strangelets, magnetic monopoles, or whatelse you have probably heared about, is scientifically confuted. It is adressed at the concerned public and you can read it here. In addition, if any of these phenomena discussed in this report could be discovered at the LHC, this would be a reason for celebrations and dishing out some Nobels, since it would experimentally confirm cool new physics and support theoretical ideas people are thinking about.
I won't discuss issues of mini black holes, strangelets, and so on. I would like to emphasize though that LHC, like any major industrial facility (LHC is not an industrial facility, but it certainly has industrial scale), has some serious safety risks:
dmckee mentioned radiation, but the beam itself is not the only source of risk; the superconducting magnets also present some significant risks, as they also contain a great amount of energy. In principle, this energy can be released during, e.g., a quench event (transition of superconductors to a normal state), producing explosions, damaging helium vessels, and so on.
It is my understanding that safety is a paramount concern at LHC. The following report covering the notorious incident at LHC may be of interest: Report on 080919 incident at LHC , in particular, the report says:
"The technical parameters of the LHC are beyond precedent, and the energy stored in the superconducting magnets huge. Consequently, operation of this machine will always comprise a certain technical risk."