Chemical propulsion is only used extremely sparingly. Since there is no air resistance in space (or, only negligible resistance due to trace gases and solar winds), the probe continues to move.
Absent any gravitational influence, the probe would continue in a straight line, forever. This is what the first three of Newtons laws tell us, especially the first.
However, gravity does act on the probe, as do solar winds etc. cetera. This makes course corrections necessary (plus, the initial burn is not perfectly precise). However, the course correction are magnitudes smaller than the initial burn to bring the probe en course.
You need about 9400m/s of delta-v to reach LEO. And according to some napkin math, about 6600m/s to reach Jupiter. For further out, you need even more. Almost all of this delta-v is provided by rocket the spacecraft is launched with. In fact, new Horizons was launched with 16.26 km/s, becoming the fastet object ever launched from earth.
For comparison: New Horizons itself has a delta-v budget of 290m/s (or 0.29km/s) of delta-v available on itself, as it only carries ~77kg of fuel. That is less than 2% of the delta-v budget. The other 98% come from the launcher.
Further velocity can be gained by gravity assists. These maneuvers require very little delta-v, but need a lot of planning and require the planets to be in the correct position - they need to be in alignment. Thus, utilizing gravity assists requires patience. This is why the Voyager missions were rushed so quickly. The alignment that offered them the opportunity to visit all outer planets except Pluto at once. This alignment only occurs every 175 years.
All this is to say that space probes do not use fuel continuously, but only very small amounts of fuel, very intermittently. Practically all velocity is gained at launch and by gravity assists, the on-board fuel is for course corrections and attitude control only.
Ion propulsion on the other hand changes this, a lot. Ion engines can in fact provide large amounts of thrust and do so over large timespans - weeks & months, often even totaling times in years (compared to minutes for chemical propulsion). The Dawn spacecraft changed it velocity by 11.49 km/s over the span of 5.9 years.
However, due to the incredible low thrust, ion propulsion is not suitable for every kind of mission. It also needs a lot of electrical energy to work. While demonstrated to work in 1964 and used since the 70s for station keeping in earth orbit, it only became used for deep space missions at the turn of the millennium, with Deep Space 1 in 1998 being the first spacecraft to use it.