If set to false, the program never checks whether the ideal $I$ or any $C_{y,I}$ or $N_{y,I}$ ideals are unmixed. Setting CheckUnmixed=>false will speed up computations since it is not performing a check of this condition but comes at the cost that not all the necessary conditions are checked. Notice that if isGVD(I, CheckUnmixed=>false) returns false, then $I$ is conclusively not geometrically vertex decomposable as there is some other condition that is not met. The default value is true.
If you know that $I$ is unmixed but want to check unmixedness for $C_{y,I}$, $N_{y,I}$, and any later ideals, use IsIdealUnmixed instead.
The following is not unmixed [SM, Example 1.6] and hence not geometrically vertex decomposable. However, if we disable the unmixedness check and skip the CohenMacaulay check, isGVD returns true.




[SM] Hero Saremi and Amir Mafi. Unmixedness and Arithmetic Properties of Matroidal Ideals. Arch. Math. 114 (2020) 299–304.
As in the above example, if you set CheckUnmixed=>false and you do not already know that both $I$ is unmixed and all later $C_{y,I}$ and $N_{y,I}$ ideals are unmixed, then the output of isGVD or any other GVD method cannot definitely conclude that $I$ is geometrically vertex decomposable, as not all of conditions in the definition were checked.
The object CheckUnmixed is a symbol.