Yrurtj

Since the gravitational force of the sun is a central acting force, the torque on an orbiting body about that center is zero. But if the sun were to hypothetically disappear, the torque would remain zero. This would mean that angular momentum of the orbiting body should be conserved, but I find everywhere that planets will move in a straight line tangentially. What is wrong in my reasoning?

The flaw in your reasoning is thinking that straight line motion at constant velocity does not constitute constant angular momentum about some point, but it actually does.

Angular momentum is given by
$$mathbf L=mathbf rtimesmathbf p$$
Without loss of generality, let's assume our object is moving along the line $y=1$ in the x-y plane, and we are looking at the angular momentum about the origin. Then our angular momentum must always be perpendicular to the x-y plane, so it will be sufficient to just look at the magnitude of the angular momentum
$$L=rpsintheta$$
where $theta$ is the angle between the position vector and the momentum vector (which is the angle between the position vector and the x-axis based on the set up above).

Now, since there are no forces acting on our object, $p$ is constant. Also, $rsintheta$ is just the constant $y=1$ value given by the line the object is moving along. Therefore, it must be that $L$ is constant.

This shows that no torque (conserved angular momentum) is not enough to uniquely determine how the motion behaves. In orbit, there is still a net force acting on our object. Without gravity, there is no net force. The motions are different.

The angular momentum of each planet is indeed conserved, but that merely means that each planet will keep on rotating around its own axis. See https://en.wikipedia.org/wiki/Rotation_period.

The planets would continue straight along the tangential line of their orbit from their position at the time of the Suns disappearance. The gravitational force exerted by the Sun would no longer be there to keep the planets in their orbits.

The planets in the Solar system are also exerting forces on each other, so the lines will not be mathematically straight, and depending on where the planets were in their orbits at the time of the Suns disappearance, a lot of different things could happen later on.

The planets will continue moving with the momentum they have at the moment the Sun disappears, apart from mutual attraction. These orbitals conserve planetary angular momentum is conserved. This is clear from the formula $L = vec r times vec p$. $r sin theta$ and $mv$ remain constant after such an event.

The presence of gravity means that the orbit has to be conical to conserve angular momentum, the ellipticity depending on a combination of L and energy. In the absence of gravity straight inertial movement does the job.