This is a bit specialized, but a common misconception in low-dimensional topology (particularly in knot theory) is that any change of basis in homology is realized by a diffeomorphism, hence (for a surface) by an action of a mapping class. I think this is exactly the type of false belief being described (I falsely believed it for a long time myself).
Common misconception: Let F be a genus 2g surface, and let $b_1,\ldots,b_{2g}$ be a primitive basis for $H_1(F)$, represented as embedded curves in F. Any change of basis for $H_1(F)$ is realized by an action of the mapping class group of F on the embedded curves.
This is rubbish- the action of the mapping class group on homology is by $Sp_{2g}(\mathbb{Z}))$, which for $g>1$ is a proper subgroup of $GL_{2g}(\mathbb{Z})$, the group of base-changes of $H_1(F)$.
As an example of what you can't do with a diffeomorphism of a surface, consider a disc with 4 bands A,B,C,D attached, so the order of the end sements is $A^+B^-A^-B^+C^+D^-C^-D^+$, together forming a surface. A basis a,b,c,d for $H_1(F)$ is given by this picture as 4 loops going through the cores of the bands A,B,C,D correspondingly. You can add a to b, b to c, c to d, or d to a by diffeomorphism of F (sliding adjacent bands over one another). However, although you can add a to c algebraically, because bands A and C are "not adjacent in F", there is no corresponding diffeomorphism of $F$.
One place this mistake manifests itself (cranking up the level of terminilogy for a second) is in thinking that unimodular congruence of a Seifert matrix corresponds to ambient isotopy of a Seifert surface.
A related common mistake (closely related to this question):
Common misconception: Any homology class is represented as a submanifold. Maybe even as an embedded submanifold.
