I accept the result of Kant's antinomy on time, from which he deduces that time is a human interpretation, imposed on thought, and not a real thing. Then to address a question like this, we have to decide what we are really talking about when we speak of time passing. For me, that is the Second Law of Thermodynamics.
If time is not a dimension, but is the effect of the ongoing increase of entropy, then it cannot be infinite in either direction, especially if it is linear.
As we reach utter pattern-lessness, time should have more and more difficulty moving forward. If there is in fact a state of such disorder that it no longer allows for the osmosis of heat, time will eventually stop. At the same time, if entropy decreases backward in time, it must lead back to a state of very high order. At either end, time slows asymptotically as either freedom or movement becomes so rare that what is going on cannot be seen as the passage of time.
In one interpretation, the fluctuation theorem says time can flow in either direction, but must reverse at a point of zero or maximal entropy. So if we managed at the right point to reverse the dependency of our processing and storage mechanisms, we could experience the time reversal as continuing forward, with all of our thermodynamics reversed.
This implies to me that the two ends of time need to be symmetrical, and it cannot be possible to experience a limitless future with a fixed beginning, in a consistent way. If time is reversible, imaginary time can be boundless in either direction, and real time as process we actually observe must stop at both ends.