There are a number of ways to annihilate a planet to consider, all of which have different effects. In addition, it depends on the interpretation of "not harming humans" both in terms of timescales (within the next year, next millenia, next million years, etc.) and in terms of harm (not killing/injuring immediately, not increasing likelihood of death or injury, etc.).
Blow it up
This leaves large fragments (=asteroids) floating around,some of which are likely to cross the Earth's orbit, with a likelihood of collision. A collision definitely falls under the category of harming humans, but in the short term, not something that is likely to happen.
It also removes a gravitational well from that planetary orbit, which had its own effects. In the case of the gas giants (especially Jupiter), they sweep up much of the asteroids and comets that could threaten the inner solar system (and thus us), providing a shield. They also have large numbers of moons whose orbits would change, possibly sending then careering off in random directions, with one or more likely to cross Earth's orbit.
So, if "not harming humans" means "not immediately harming humans", then anything, other than the moon, is possible. (The moon, tidally locked to the earth, performs several vital functions, one of which is keeping the earth steady on its axis. If it were to be removed, earth's rotation would become chaotic (like that of Mars) and our current, predictable seasons would eventually vary wildly.) If it means "not in the next hundred years" then I would add Venus, Mars and the gas giants to that list of no-goes (due to the likelihood of creating a large number of earth-crossing asteroids). If it means "not significantly changing the risk to humanity in the next million years" then probably only Pluto or any of the Kuiper Belt or Oort Cloud objects would be feasible to blow up.
Reduce it to insignificant pieces
The main effect of this would probably be the loss of gravitational well, but depending on how spread out the particles are, that might not be significant, and they would coalesce back into a (slightly smaller) planet over the period of a few million years.
If the particles are widely spread out, we would have a problem with loss of gravity well.
Vaporise it (convert to energy).
I'm not sure what the minimum safe distance for watching the mass-conversion of a planet might be, but it is likely to be quite far. Without doing the sums, I would hazard a guess that only the smaller asteroids and anything in the Kuiper Belt or Oort Cloud would be suitable for us to survive the blast of radiation.
It would, however, be very flashy!
The loss of the gravity well would probably not be a problem with those small distant objects.
If we assume that Jupiter was vaporised, and that it produced the same amount of energy as the fusion process in the sun (0.7% mass conversion into energy) it would produce 1.2x1042 J. For comparison, the Sun puts out an average of 3.846x1026 W, so if the explosion occurred over 10 seconds, it would be about 3x1014 times more energetic than the Sun, and outshine the entire galaxy (about 100 billion (=111 stars). If we assume that Jupiter was on the opposite side of the Sun from the Earth, it would be 968 million km away, and the energy flux would be about 1016 W/m2 at the Earth's distance, lasting 10 seconds.
A similar calculation for Neptune results in an energy flux at the Earth's distance of 4.6x1013 W/m2.
Now, the Sun is very opaque to radiation (it takes about 4 million years for photons to make their way out of the core), but I don't think it's going to be sufficiently opaque.
By comparison, a Type II supernova puts out about 1046 J over a period of about 10 seconds. A Type II supernova occurring 8 parsecs from the Earth might destroy more than half the ozone layer, producing an average flux of 3.05x1010 W/m2 at the Earth, so vaporising Jupiter would produce about 330,00 times more radiation on the Earth than the supernova. Neptune would produce about 1500 times more radiation. I don't think that any of the gas giants would be a good choice to vaporise, even on the opposite side of the sun.
Convert into a black hole
(EDIT: I've done some very rough calculations, using http://xaonon.dyndns.org/hawking/, and come to a different conclusion than before.)
This would be rather dull, as the planet disappears into a tiny black hole. For example, a Jupiter-mass black hole would be 2.8m in radius, and be barely detectable as its temperature would be barely be above absolute zero. It would evaporate too slowly to notice (well in excess of the lifetime of the universe so far at 1.82x1058 years. (I'd previously thought that it would evaporate soon after forming in a flash of gamma rays, but apparently that's only true for really tiny black holes which are also a lot hotter.)
As a result, just about any planetary object can be safely converted into a black hole; either they're too large to noticeably decay, or they're too small to put out the kind of energy that might harm us.
The loss of the gravity well would not be a problem with those objects; the larger ones with a noticeable gravity well would survive for a very long time, and the smaller ones wouldn't have a noticeable gravity well anyway.