What happens when you put water under intense pressure? Pretend you have an indestructible tube that cannot leak, inside which is water. Imagine that in each side of the tube, you have very powerful pistons
What would happen if you compress the water inside?
Would it turn into heat and escape the tube?
Would the water turn into solid because the water molecules are so close to each other?
Would the water turn into a black hole?
What would happen? 
 A: What you're asking about is usually shown in a phase diagram. The diagram shows how the "phase", i.e. liquid, gas, or one of various solid phases, exists at different temperatures and pressures:

If your cylinder starts at say $20{}^{\circ}\mathrm{C}$ and atmospheric pressure, it'll be in $\color{green}{\textbf{Liquid}}$ right near the center of the diagram.  If you raise the pressure keeping the temperature constant, it'll switch to $\color{blue}{\textbf{Ice VI}}$ at about 1GPa, or about 10,000 atmospheres of pressure: it's hard to turn water to ice by compressing it; the water at the bottom of the ocean is still water.
As you keep raising the pressure further, keeping the temperature constant, it'll go through more and more compact forms of solid ice (the diagram doesn't show "black hole", as that would be many, many orders of magnitude off the top, and can't be physically reached).
I stress "keeping the temperature constant" because (a) that's something your experiment will have to choose to do or not do and (b) because it makes it much easier to read the diagram.  The compression is adding energy to the water, from the work done by the pistons.  If you go slow, and the cylinder isn't insulated, etc, that energy will conduct away as the cylinder naturally stays the temperature of its environment. If you go fast, or the cylinder is insulated, the temperature will rise and the water will tend to go up-and-right in the diagram: You'll hit the transitions at different points.
A: Short version


*

*Around ${10}^{5}\mathrm{Pa}$, which is about normal atmospheric pressure, water's liquid.

*Around ${10}^{9}\mathrm{Pa}$, water's compressed into ice.  Ice configuration varies as pressure increases.

*Around ${10}^{12}\mathrm{Pa}$, the ice metallizes.  It's no longer a bunch of H2O molecules, but rather a soup of H and O atoms.

*Around ${10}^{16}\mathrm{Pa}$, very roughly, nuclear reactions might start to become appreciable (though not exactly common).

*Around ${10}^{31}\mathrm{Pa}$, pressures approach what's found in a neutron star.  It's no longer meaningful to speak of individual atoms.

*At some point, it probably collapses into a black hole, or something.

*Past that, it'd be rampant speculation.

Long version
Roughly, works kinda like this:


*

*Start at a typical temperature/pressure; let's say $20{}^{\circ}\mathrm{C}$ and $1{\cdot}{10}^{5}\mathrm{Pa}~\left(\approx1~\mathrm{atm}\right)$.

*Compress, assuming that the cylinder is in a heat bath of constant temperature.


*

*Another common option is to consider an adiabatic compression, in which heat is trapped too, such that temperature'll tend to increase as it's compressed.  Doing isothermal instead because it's lazier.


*As pressure increases, a few minor effects will happen:


*

*Water'll lose some volume (though it's not too compressible).

*Heat'll get generated (though it'll be lost to the heat bath).

*The chemical equilibrium will shift a bit.


*Around $P{\approx}1{\cdot}{10}^{9}\mathrm{Pa}$, the liquid water will start to compress into ice.


*

*Specifically, the water molecules will arrange themselves into an Ice VI pattern.


*Around $P{\approx}2{\cdot}{10}^{9}\mathrm{Pa}$, the Ice VI pattern might start to lose way to Ice VII.

*Around $P{\approx}6{\cdot}{10}^{9}\mathrm{Pa}$, the Ice VII pattern might start to lose way to Ice X.

*Beyond this point, things start to get speculative as we're outside the realm of experimental verification.
The plot in @BobJacobsen's answer shows a prediction of a variation of Ice XI, as does this slightly extended plot:


*As pressures increase further, it stops being "water", but instead a metal composed of the atoms that used to be in the water.  The difference is that, pre-metallization, water is H2O; after metallization, it's no longer meaningful to speak of individual molecules of H2O, much like it's not meaningful to speak of "molecules" of typical metals.
Exactly when and how metallization happens is controversial.  There're competing claims for when hydrogen metallizes, and I found a claim that oxygen metallizes relatively soon too, and then another claim that water would metallize at another pressure:


*

*$P{\approx}2.5{\cdot}{10}^{10}\mathrm{Pa}$: Early prediction of hydrogen's metallization.

*$P{\approx}9.6{\cdot}{10}^{10}\mathrm{Pa}$:  Wikipedia cites oxygen going metallic.

*$P{\approx}4.95{\cdot}{10}^{11}\mathrm{Pa}$:  There's a contested experimental claim to having observed metallic hydrogen.

*$P{\approx}1.55{\cdot}{10}^{12}\mathrm{Pa}$:  Claim for water (so both hydrogen and oxygen together) going metallic:

Based on density functional calculations we predict water ice to attain two new crystal structures with Pbca and Cmcm symmetry at 7.6 and 15.5 Mbar, respectively. The known high pressure ice phases VII, VIII, X, and Pbcm as well as the Pbca phase are all insulating and composed of two interpenetrating hydrogen bonded networks, but the Cmcm structure is metallic and consists of corrugated sheets of H and O atoms. The H atoms are squeezed into octahedral positions between next-nearest O atoms while they occupy tetrahedral positions between nearest O atoms in the ice X, Pbcm, and Pbca phases.
-"New Phases of Water Ice Predicted at Megabar Pressures" [formatting omitted]



*At some pretty ambiguous point, nuclear reactions'll probably start to become significant, likely with hydrogen's going to helium, etc..  Since nuclear reactions have started, we're outside the realm of chemistry.

*Eventually, the matter'll probably become degenerate.  Naively, I kinda picture something like a neutron star, where increasing the pressure would be like digging deeper into a neutron star's layers:

But, obviously, we're deep into the realm of speculation at this point.

*Eventually the degeneracy pressure is overcome, and it'll form a black hole, maybe.  Perhaps it'll look like a fuzzball, maybe.  Don't really know.

Discussion on high-pressure ice
For the extreme range of pressures in which water is still "water", this source seems to have a good discussion:

Very high-pressure ices including superionic ice
The state of ice at the very high pressures above ice X has only recently been reached experimentally. Modeling gives a confusion of possibilities. As such modeling, but of lower pressure ices, does not give accurate results as compared with experimental structure information, it is expected that these results are, at best, indicative. Density functional calculations [1709] indicate a pressure-induced initial displacement of the ice-ten atomic layers to give an orthorhombic Pbcm structure. At higher pressure, this may be followed by the squeezing of the H-atoms from their midpoints to give a Pbca structure and then, at over a terapascal (TPa, 107 atm), to a metallic ice, consisting of corrugated sheets of H and O atoms with the H-atoms at the octahedral midpoints between next-nearest oxygen atoms [1709]. Alternative views have been given; one is that the orthorhombic Pbcm structure is superseded by a _Pmc_21 phase above 930 GPa, followed by a _P_21 crystal structure at about 1.3 TPa and finally the metallic C2∕m phase above about 4.8 TPa [1818]. Another study shows that trigonal P3121 and orthorhombic Pcca phases become stable in the ranges 0.77-1.44 TPa and 1.44-1.93 TPa [2114] respectively. Such ices are not molecular and may be considered as protons and oxygen dianions with mobile electrons [1666] and are expected at the core of giant planets such as Jupiter and Saturn A partially ionic phase consisting of alternate layers of OH- and H3O+ at low temperatures has been suggested [1810]. Several new phases may convert into one where the coordination number of oxygen increases from 4 to 5 with a significant increase of density [1818]. At pressures over about 5 TPa, it has been suggested that a phase splitting occurs with (the components of) H2O decomposing into a cubic Pa-3 H2O2 -formula phase and a hydrogen-rich phase, with metallization predicted at a higher pressure of just over 6 TPa [2114].
A new superionic phase was proposed with an approximate triple point of about 1000 K, 40 GPa with liquid (supercritical and ionized) water and ice-seven at high temperatures (~1500K) [1572]. In this phase, the hydrogen ions (protons) were expected to be highly mobile, behaving like a liquid, and moving within the solid lattice of oxygen ions.The recent experimental discovery of superionic ice has reinforced this prediction [3199]. Using shock compression of ice-seven, it was shown that ice melts near 5000 K at 190 GPa.
-"Water Structure and Science" [links omitted; formatting partially reproduced]

