What happens to water under high pressures without possibility of escape?
It depends on what you mean for high pressure and on the temperature, however water phase diagram can help you to understand what will happen. This is from wikipedia User Cmglee:
You can see that at high pressure water assumes a solid form (ice): you will have ice X at 100 GPa and ice XI at 1 TPa (temperature range 0° to 650 °K) these sort of ices have different lattice and internal energies.
What would happen to water as pressures increase towards infinity without possibility of the water escaping confinement?
Unfortunately chemists don't deal with infinity, basically however there will be a point where the water molecule wont longer exist all the bonds will brake. After that probably you will have something called electron-degenerate matter have a look here, here is all about quantum mechanics, Pauli exclusion principle and black holes so on more Physic S.E. stuff.
Could triple points in water play a role in keeping temperatures low under high pressures via fluid thermodynamics?
You can also note that there are others triple points like (100 k, 62GPa) this however don't directly affect the properties of water but are the properties of water that determine where are the triple points. In a close system kinetic energy is constant so I think that fluid thermodynamics doesn't matter in this case is more related to equilibria.
More info about water phase diagrams here.