Two-dimensional BF theory is an example of a topological gauge theory, that is, a gauge theory whose correlators do not depend on any geometric data. Imposing the Lorenz gauge-fixing condition introduces an auxiliary geometric data in the form of a metric. However, we will show that the theory becomes topological conformal, i.e. it depends only on the conformal structure of the introduced metric. Moreover, the stress-energy tensor is Q-exact (hence vanishes in Q-cohomology and therefore on physical states). Going beyond gauge-invariant (i.e. Q-closed) observables, allows one to define interesting structures such as topological correlation functions, a BV algebra structure on the Q-cohomology and a toy version of Gromov-Witten theory.