Are you using prewrap? (sort of thin foamy stuff) It kind of sticks to itself and then the tape sticks to it. It provides somewhat of a barrier against sweat. I had to get taped for track all the time in a very hot climate and I never really had problems. Maybe because it was my ankle area not my knees.

Something else is the flexible bandage stuff...it has a bunch of names but I'll try to describe it. I call it vetwrap because it's just like stuff I'd use on horses to keep gauze pads on an ankle and stuff. It's this stretchy bandage that looks like a thin Ace bandage (comes in different widths). It is adhesive but doesn't stick to your skin, just itself. I'd still recommend tucking the end under, though. It's not as good against sweat as prewrap and white cloth tape, but it's flexible unlike the white tape, so maybe it'll move a little with you instead of slide around? I use it usually to protect scrapes, but I have taped my ankles with it a few times, and it's been fine.

Be sure you're using enough tape too to really secure the wraps.