What happens to water under high pressures without the possibility of escape?
It depends on what you mean for high pressure and on the temperature, however, the 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 what is labelled as hexagonal ice XI in the diagram but is actually superionic water or ice XVIII at 1 TPa (temperature range 0 to 650 K). These sorts of ices have different lattice and internal energies.
What would happen to water as pressures increase towards infinity without the possibility of the water escaping confinement?
Unfortunately, chemists don't deal with infinity, basically, however, there will be a point where the water molecule won't longer exist all the bonds will break. 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 other triple points like (100 k, 62GPa) this however doesn'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.