You're not doing anything wrong. Once dried, gummed brown tape won't come off easily. I usually don't bother trying, but instead I cut off the edges of the sheet or, if it's to be frame, know it'll be hidden a card mount and leave most of it.
To remove it, you'll need to dampen the paper tape again, so the glue is moistened; then it'll lift off again. Use a damp cloth or sponge, but be careful not to let water run onto the painting or it'll spoil it. I'm not sure the risk of ruining the painting is worthwhile.
You can avoid stretching watercolor paper by using heavy-weight paper which won't buckle when it gets wet (unless you totally soak it). But this paper is far more expensive.