Three conditions must be fulfilled:
The potentially burned lens is in the focal plane of the other lens;
The other lens is big enough to gather enough solar energy; and
The potentially burned lens is absorbing enough that enough of the the light focussed on is dissipated as heat within it, such that the lens temperature rises either enough to denature plastic / melt glass or set up a differential temperature profile and large enough attendant internal stresses to shivver the lens, as appropriate to the situation on hand.
Are you destructively testing lenses, or are you wanting to ensure a solar system doesn't fry itself?
I'm guessing the aim of your question is the latter. Most lens systems do not have real focusses inside them so this kind of thing is not often a problem, but a notable exception is the Keplerian Telescope and like systems, which must form an intermediate focus inside the lens system so as to put the exit pupil at the viewer's eye / physical output of the device: this kind of arrangement might be used in a relay lens system where you want to confine light into a tight, long tube. Certain endoscopes for example uses this concept.
Bottom line: you must raytrace throughout your system, and, if there are any regions where the beam is squeezed into a tight region whilst propagating through a physical lens element, then you must calculate the rate of dissipation of light into that element, and do a thermal conduction / convection analysis to check that the temperatures reached are not high enough to damage the lens in question.