It is not clear what Vijayan, Banerjee and Singh intend to do to "not implement" the law. One part of the bill adds a paragraph
6B. (1) The Central Government or an authority specified by it in
this behalf may... grant a certificate of registration
These states may decide to not cooperate. This would be analogous to US jurisdictions refusing to cooperate with federal authorities attempting to enforce immigration laws. However, the Bureau of Immigration (analogous to US ICE) may have sufficient resources to enforce the law without any state aid.
The other possibility (suggested by Banerjee's statement that W. Bengal will not allow the government to "throw out a legal citizen of the country or turn the person into a refugee")
is that this relates to the theory that the law will be used to expel Muslim refugees. A state could not directly prevent the central government from carrying out such a plan, but it could refuse to cooperate. Since as I understand it public order, police, and prisons are on the State List under Art. 246 of the Constitution, and the Union list do not seem to provide for the kind of sweeps that have been suggested could result from the law (unless the military executes the plan). There is no question that citizenship laws are on the Union List.