There was an interesting conversation at the RTCUK Sleepover about DEET maybe degrading climbing ropes. It was scary enough to make you think twice about risking Lyme's or a nasty fall. After all, as a solvent DEET will melt plastic and denature your sunglasses, camera and so on.
I couldn't find anything definitive online, but everything you can find points to this quote from a report which now seems to be unavailable:
Personally, I'm taking this as permission to not worry. The most up-to-date forum discussion about this I could find was this one, and there seem to be no accounts online of rope failure after DEET contact. Your mileage may vary, but for me avoiding the relative certainty of again having to extract gangs of voracious parasites from my nethers beats the theoretical risk of equipment failure.