Particles that are already entangled can be brought to large distances from one another and remain entangled. This means that measurements on each of the two particles will show statistical correlations (IE my result here has a statistical relation to what you chose to measure over their) that one can prove would not be possible with a so-called "local hidden variable model". These statistical trends are not enough to send signals faster than light.
As for getting the particles entangled in the first place, I am not aware of any forces that can occur at a distance without an exchange particle (indeed if their were you COULD send signals faster than light). So their should be no physical Hamiltonian that is nonlocal (acts on things at two places at once).
Their might be some Hamiltonains that people use in models that appear non-local (the Ising model Hamiltonian for instance: https://en.wikipedia.org/wiki/Ising_model, has an interaction between adjacent spins), but these will always be approximations. For example the true Hamiltonian for the spin-spin force should include the exchange photon that propagates between the spins.
Add on later:
It is actually possible to entangle two particles that have never interacted with one another (even indirectly). If particle A is entangled with particle B (A-B) and particles C and D are entangled (C-D), it is possible to perform a procedure called "entanglement swapping" where you get particles B and D, perform some process, and arrive in the state where A and C are entangled (A-C).
This can be done even if A and C were never anywhere near eachother, and even if they have never interacted with one another (even indirectly).
This was the approach used in: https://www.nature.com/articles/nature15759