If I have a C++ class which contains a static member variable, does the accessor method for this variable need to be static as well? Also, are there any issues that might occur if I inline this method?
It doesn't need to be static, but unless it's doing something specific to a particular instance of the class, there's no real reason not to make it static anyway.
This shouldn't effect inlining in any way.
The whole purpose of writing an accessor method is to hide the implementation of certain property of the class from external users. What you want to hide and what you don't want to hide is something only you can decide. Decision like this cannot be made mechanically, as in "since the data member is static, the accessor should also be static". This is a hopelessly flawed approach.
Again, the point of writing the accessor is to decouple the user from all (or some of the) knowledge about the underlying data member. The data member might not even physically exist. The user does not need-to-know whether it physically exists or not. The data member itself might physically exist today and disappear tomorrow. The user is not supposed to care about that. This is what you achieve by forcing users to use an accessor.
So, in your case, by making the accessor static you automatically declare and expose the fact that the corresponding property is specific to the entire class, not to a particular object of the class. Note, again, that the static-ness of the data member is beside the point here: no one will ever know or care whether there's a physical data member behind that accessor.
By making the accessor non-static you automatically declare and expose the fact that the property might be specific to a particular object of the class.
This is what should guide your decision. You don't provide enough details about the property in question, so we can't help you here to decide.