Here are four tempting fallacies that I have seen, that are in my opinion are all teachable and interesting:
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.
If you lengthen all three edges of a triangle, its area increases.
If $F$ is a field with two finite-index subfields $K$ and $L$, then $K \cap L$ also has finite index in $F$.
There are exactly two Lie groups up to isomorphism that are diffeomorphic to a pair of circles.