They might escape from the solar system, if the angles are right. If not, they'll probably wind up in elliptical orbits around the Sun. We'll use a simplifying assumption that the orbits are circular to make the calculations easier; All objects mentioned have orbital eccentricity of less than 0.05.
The mean orbital velocity of Jupiter is 13.1 km/s, which makes Solar escape velocity at Jupiter's mean distance from the sun 18.5 km/s, a difference of 5.4 km/s.
The mean orbital velocities of the Galilean Moons of Jupiter are:
- Io: 17.3 km/s
- Europa: 14.3 km/s
- Ganymede: 10.9km/s
- Callisto: 8.2 km/s.
Which means, if the geometry is right, with the direction of travel of the moon in the same direction of the direction of travel of Jupiter at the time of the disappearance, the resulting vectors could add up to any of the Galilean moons exceeding solar escape velocity if Jupiter suddenly vanishes, and exiting the solar system.
Doing some trigonometry to find the limiting angle:
- Io: Reaches Solar escape velocity if the moon was traveling within 106° of Jupiter's direction.
- Europa: Escapes if traveling within 95°
- Ganymede: Escapes if direction within 79°
- Callisto: Escapes if direction within 62°
There's also a tiny, tiny chance of Europa or Io being in the right positions and having the right direction to hit the Sun if their directions at the time of disappearance were exactly right, but the chance of that is extremely unlikely.
For comparison, Saturn has a mean velocity of 9.5 km/s, (Solar Escape Velocity at that distance is about 13.4 km/s) and Titan has a mean velocity relative to Saturn of about 5.5 km/s.
Titan could escape under the right conditions (Within 56° of Saturn's direction of travel), but doesn't have enough Saturn-relative velocity to have a chance of crashing into the Sun without help.