Does evaporated hand sanitizer pollute the environment?

Assuming you are talking about ethanol: it is a common chemical substance, generated naturally from e.g. rotting fruits. So any pollution from hand sanitizers should be negligible.

What happens to the alcohol in the atmosphere is an interesting sequence of reactions. I took the info below from an article (see link below).

In the air, ethanol is oxidized quickly (half-life is a few days) into acetaldehyde $CH_3CHO$, which is further oxidized into other compounds, most of which are harmless. However, if the air contains nitrogen dioxide $NO_2$, it can form peroxyacetyl nitrate $CH_3C(O)OONO_2$, which is dangerous.

Source