What would happen if you put your hand in front of the 7 TeV beam at LHC?
Well, there is the unbelievable story about a guy who actually put his head in a proton beam, the Russian scientist Anatoli Petrovich Bugorski. This happened at the U-70 synchrotron, near Moscow at the Institute for High Energy Physics.
But the thing is, is that he actually didn't feel any pain. He did suffer from epyleptic attacks and damage to his skin and brain. Citing from:
What would happen...?
Bugorski was leaning over the piece of equipment when he stuck his head in the part through which the proton beam was running. Reportedly, he saw a flash “brighter than a thousand suns”, but did not feel any pain. The beam measured about 2000 gray when it entered Bugorski’s skull, and about 3000 gray when it exited after colliding with the inside of his head.
To be clear, a gray is the unit of absorbed energy, and 1 gray equals 1 Joule/kg. For comparison, CT scans work in the ~50 mGy range and radiation therapy in the ~50 Gy range. The bizarre thing is, if you're exposed to >5 Gy it's usually lethal. But that applies to usual radiation, gamma radiation. The effect of a proton beam is less known.
The protons in the beam can be accelerated up to ~70 GeV, which is 1/2000 of the LHC range. There's also the beam luminosity, which is a measure for the number of particles flying through a unit area per unit time. The LHC can reach 10^34 1/(cm^2 s^2) and the U-70 about 10^32 1/(cm^2 s^2). So the most naive estimate would be an increase of 10^3 - 10^5 of the radiation dose -- but that's more like an upper bound.
It won't destroy your hand or blast it off. But the effects of radiation are quite severe, and you will end up with a mutant hand.
A mis-steered beam at CEBAF simply cut a hole thought the niobium wall of the cavity and flooded half the accelerator with helium (super-conducting cavities need a liquid helium jacket to work...).
We were down for more than a week.
That is an electron beam machine, and very high current (up to $400\,\mathrm{\mu A}$!), so the details would be rather different than the LHC beam.
Likewise, about $60\,\mathrm{\mu A}$ of $5.5 \,\mathrm{GeV}$ beams from that machine partially melted one of my iron targets (thick enough that about 6% of the beam interacted with the target) despite a raster spreading the beam over roughly $2\,\mathrm{mm}^2$, because we didn't have good enough thermal contact with the water-cooled frame of the target ladder.
High energy, high current beams can carry a lot of power.
Back to the question as asked:
Treat your hand as water. The particle data booklet puts the energy loss per proton at around $2.5\,\mathrm{MeV/g/cm}^2$ or something like $4$–$8\,\mathrm{MeV}$ though your hand, depending on how chubby you are. (We're only a few orders of magnitude above the minimum ionization energy, so this is not very sensitive to the actual beam energy.)
Phillip says $1.2 \times 10^{11}$ protons per beam in the ring, about 11,000 passes per second ($3.0 \times 10^8\,\mathrm{m/s} / 27\,\mathrm{km}$), so $1.3 \times 10^{15}$ protons per second is $8 \times 10^{15}\,\mathrm{MeV/s} = 1300\,\mathrm{J/s}$ is a fair bit of heat, and results in heating of about $300\,\text{K/s/(cubic cm exposed)}$. To finish up here we will have to know something about the beam diameter.
Your reflex time to move your hand is on order of $0.1$–$0.2$ seconds.
It's going to hurt: you will get badly burned, and the damage will extend though the whole depth of the exposed flesh, rather than being limited to the surface as with the contact burns we are all familiar with.
Each bunch of protons circulating in the LHC at full power will have about 120 billion protons of 14TeV each, i.e. $1.6 \times 10^{24}$ eV which is about 250000 Joules. Compare that with a high powered rifle bullet carrying 1000 Joules or a small grenade that releases 600000 Joules.
But how much of that energy will be released in your hand? It is said that the protons at the LHC can penetrate 10m in copper so many will go through 2cm of flesh without losing much energy. This means the destructive energy released in the hand could be just 1/1000 of the total in the beam, so it might be like being hit with a small bullet.
I think it will still be enough to do a lot of damage. It will also be in the form of high energy radiation which is going to spread out from the impact point. It's difficult to be sure how much energy will convert to different forms such as heat and blast, and in which directions, but whatever way it goes I think there will be a lot of initial damage and the radiation after effects will also have the potential to be fatal.
One last edit: It might be worth mentioning that they can circulate up to 2800 of these bunches at once, so it might be more like being hit by 2800 small bullets in a fraction of a second.
Edit: question was changed from 14TeV to 7TeV for correct LHC beam energy, so use half the numbers above