NASA has approved the New Horizons mission to 2014MU69 on 1 January 2019 that will require operations at least into 2021 to download the data. However, after that, things get a bit fuzzy - New Horizons will still have a substantial amount of fuel left, so it could theoretically visit another KBO - if one was found close to its trajectory. Even if another object in the Kuiper Belt doesn't turn up to visit, NH is on a trajectory that will take it out of the Solar system.
The Voyager probes are currently (4 September 2017) traveling 17kps and 15.4kps away from the Sun, in interstellar space (beyond the heliopause), and transmitting scientific data. Although it was the fastest vehicle ever launched, New Horizons is "only" going 14.2kps, so it will take quite a bit longer to reach the heliopause than the Voyagers did - but it will get there, and go beyond.
Assuming it doesn't have a fatal crash within the Kuiper Belt, how long could New Horizons continue to return scientific data? Is it limited by its data rate at distance, or by the power available in its RTGs?