Cambered airfoils generate induced drag because they have a pressure differential between the upper and lower surface. However symmetrical airfoils don't. So does this mean that symmetrical airfoils have no induced drag?
If an airfoil is producing lift, then it will be producing induced drag.
Both cambered and symmetrical airfoils have an angle of attack at which they produce no lift, no induced drag, and no pressure difference between top and bottom. While this angle lines up nicely with the geometrical mid line of the symmetrical airfoil, it is offset from the apparent mid line of the cambered airfoil.
The cambered airfoil can produce more lift before the stall in the 'normal' lift direction, which is why it tends to be used for surfaces loaded in only one direction, like wings. A symmetrical airfoil would be used for control surfaces which might see equal loading in either direction.
Mmh, here there is a MISUNDERSTANDING OF TERMINOLOGY and even the wiki.en page about induced drag is a bit misleading.
Induced drag is a strictly 3D phenomenon, so speaking about induced drag for an airfoil is physically and mathematically not correct since airfoils are, by definition, 2D geometric shapes.
An airfoil generates lift, drag and moment, even if it is a simple 2D shape. But what is defined as "induced drag" is an additional contribution to the drag that appears only introducing the third dimension (it is a bit like when the flow becomes supersonic, then yet another source of drag - the wave drag - appears as well). So, the exact geometry of the 2D airfoil is not important here, only the fact that the "wing ends" is involved in generating the induced drag.
Qualitatively the induced drag can be explained looking at what happen at the wing's tips i.e. when the wing ends. On the upper surface of the wing there is an underpressure while on the lower part of the wing there is an overpressure. At the tip, the underpressure over the wing sucks in the air which is pushed away from beneath the wing and "escapes" around the wingtips. This generates behind the wingtip a typical vortex as visible in the following picture:
B-727 in flight during vortex study with wingtip smoke generators. Source: https://www.dfrc.nasa.gov/Gallery/Photo/B-727/Large/ECN-3831.jpg
More correctly, a "sheet of vortex" is released from the trailing edge of the entire wing as visible in this standard representation:
Vortex sheet in the wake of a wing of finite span. Source: https://cdn-images-1.medium.com/max/1600/0*2gL-QOXqx51qNdAE.jpg
But normally only the vortex at the tip is taken into account for the sake of simplicity (even if this is not really correct). This vortex sheet locally decreases the angle of incidence therefore reducing the total lift generated by the wing. To compensate for this reduction, the angle of incidence has to be increased with a related increase of drag. This latter, and only this, is the induced drag. I perfectly agree that the naming is a bit misleading, but this, and only this, is the definition of induced drag.
Would you believe I have a draft journal article about this. The use of the term "aerofoil" is a clear implication of 2-dimensional flow, or what Hurt  (and many others) call and infinite wing (AR = $\infty$). The tern wing is a clear implication for 3-dimensional flow. There is a clear logical distinction between how we teach these in aerospace (aeronautical) engineering and is best captured by Anderson . That is, we teach 2D aerodynamics first, then we teach 3D aerodynamics.
Now, the aforementioned journal article I am working on is about confusions with this. In that, I have studied education literature on lift, and only one third of the time when an "aerofoil" is used and the implication is 2D flow, do they actually get this correct. For wings and 3D flow, this is much better, it is three quarters of the time when they are talking about wings and 3D flow that it is correct. So, as is reflected here in this discussion, there is a misunderstanding about the fundamentals of 2D flow.
As has been pointed out by others, there is no induced drag in 2D flow. Hence, the answer to the question is no, symmetric aerofoils, like all aerofoils, do not produce induce drag. Induce drag results from 3D effects, whereby energy is wasted by a wing imparting a downward velocity component to the air. This image from Hurt  is the best reference to this and is sadly never referred to once in 105 articles about lift education (or in books on the topic). As such, I recommend reading Hurt for the best description of induced drag. You will note in the image that the vertical velocity component for 2D flow returns to zero, while for 3D flow, it does not. Happy to share either some potential flow of CFD sims that support what was clearly well known in the 1960's.
Induced drag is just the component of any aerodynamic force which lies in the direction the aircraft or airfoil is traveling in (aligned with the relative wind). So of course it does. If it is producing any aerodynamic force, then there must be drag. Only if we arbitrarily separate the total drag into multiple pieces and call one of the pieces induced drag can we make this distinction. How we use the words sometimes makes things unnecessarily confusing. Remember, we used to describe the forces generated by the airflow in the cavity between the F-14 engine nacelles as "Lift". Does that Lift generate Induced Drag? How about the Supersonic Shock Wave Lift generated by the XB-70 when it lowered the wingtips? Much confusion is generated by using definitions designed to be useful in one context in areas where they are not applicable.
It could, (and doubtless will), be argued, (strictly based on arbitrary semantic definition), that unless there is Lift, there cannot be induced drag, but that argument is based on semantics, (that induced drag, by it's own name, is the drag induced by, or due to Lift). That definition however, although useful at times, is at best misleading. It ignores the fact that Lift itself is a component of the same aerodynamic force - the component that lies normal to (perpendicular) to the aircraft/airfoil direction of motion or relative wind. Lift AND Induced drag are both just components of any aerodynamic force. One is not the cause of the other. Going strictly by this interpretation, If there is no "Lift", (because the total Aerodynamic force is itself aligned perfectly with the aircraft velocity vector), then it doesn't make sense to call it Induced Drag. It is, however, still there, and it is still the component of the force aligned with the velocity.