I bought a trainer at the very cheap end of the spectrum. As we get what we paid for, I opened the box and it came with no skewer...
My bike has skewers with plastic ends (how hard is that plastic is something I don't know), so I'm worried that the trainer can damage my skewers somehow. This answer says it does, but it doesn't explain how (or if I can keep using the bike on the road with the skewer's top damaged). The other answer in the same Q/A says "cosmetic damage".
Here's my skewer:
And this is where it fits in the trainer (if that's relevant):
Yes. QR skewers are not designed to carry the weight of the bike. When riding, the weight is transferred from the axle to the dropouts directly, with the skewer just holding the friction interface solid between the two. A standard skewer can be damaged by using it on a "wheel-on" trainer.
What you are looking for is a trainer skewer, which is a drop-in replacement for the regular skewer, has a stronger material and steel endcaps for the trainer to grab onto. Your Local Bike Shop should carry a selection.
To elaborate a bit on the other answer, your current's skewer's nut is plastic. I assume that the skewer's lever is also plastic. Clamping the skewer inside the trainer will scratch the plastic on both ends.
The skewer clamps the wheel and dropouts laterally. On the side with the handle, there's a cam that translates the force you used to close the lever into clamping force. I don't think that a bit of cosmetic damage will be fatal to the skewer. In particular, you would have to crack the drive-side nut through or damage the cam to jeopardize the skewer. I think the issue is that over time with repeated use, you could risk this happening to the hard plastic parts.