Here are four tempting fallacies that I have seen, that are in my opinion are all teachable and interesting:

 1. A finite covering space of a disk with finitely many holes, is again a disk with finitely many holes; in particular it is still planar.

 2. If you lengthen all three edges of a triangle, its area increases.

 3. If $F$ is a field with two finite-index subfields $K$ and $L$, then $K \cap L$ also has finite index in $F$.

 4. There are exactly two Lie groups up to isomorphism that are diffeomorphic to a pair of circles.