The entanglement of the two states changes. From the perspective of the exterior observer the particle which falls into the black hole is redshifted and time dilated according to the delay coordinate
τ = r - 2m ln(1 - 2m/r)
so that it is never observed to reach the horizon, and only does so as τ -- > -∞. Let Alice be the exterior observer and Bob an observer who falls in with the other EPR pair. Assume Alice performs a measurement and sends the result by a classical signal to Bob. Bob uses this information to reconstruct the entangled state with some ancillary state. If Bob is on an appropriate geodesic where he can receive the classical signal then the teleportation can take place. However, if Bob makes the measurement any signal he tries to send back out to Alice will never reach her. The teleportation fails. Bob may communicate this classical signal at any point just above the event horizon, where if Alice has a receiver with a long enough wavelength band she can reconstruct the state. If Bob send the signal after passing the horizon then Alice will wait until the black hole quantum decays and never receive it, or never receives it in a form recognizable.
The situation does become a bit subtle however. The entanglement between the EPR pairs becomes transferred to an entanglement with the black hole. From the perspective of Alice the quantum information of Bob and his EPR pair becomes Lorentz contracted and as increasingly UV frequencies of particles are apparent there is an increase in the number of "partons" which appear excited. This means the transverse direction of this information, or equivalently the length of strings, become elongated and spread across the horizon. Eventually at the stretched horizon r = 2m L_s (L_s = string length) everything is merged into one great string at the Hagedorn temperature, but which is redshifted to near zero temperature as seen by Alice. The EPR entanglement is lost and the entanglement transferred to the black hole. However, as the black hole quantum radiates the modes on this "great string" are demolished and states or string appear outside the black hole. So the signal that Bob sends does emerge from the black hole, but in some encrypted form. If Alice as a quantum recipe of everything which made the black hole she could then in principle reconstruct everything which went into the black hole, including Bob and his EPR pair, and the entanglement between Bob's state and the black hole transferred once again to the original EPR pair and the teleportation then completed. However, Alice must have this information, or else it is like having the winning number on a lottery ticket that you threw away or lost.