I have experience with surgically implanting radio telemetry tags in the abdomen of different species of fish, they're made of metal encased in a cylindrical tube of clear hard plastic. Multiple times I've taken them out of fish (living and dead) that have had them in for a year or two. What I've found is that their bodies actually absorb the object and begin calcifiing and attaching flesh to the object. I have no idea if that's the same with humans but it happens in fish.
One thing that is missing from the responses so far is that these objects can (and typically do) result in a low grade chronic inflammatory response that typically results in the formation of a fibrous capsule surrounding the implant. Also if particulates break off or form with the degradation of the material than it can result in macrophages being unable to destroy that particulate matter and so they fuse with other macrophages resulting in the formation of giant cells to try and destroy it resulting in frustrated phagocytosis.
As others have said certain materials are better and are used a lot as implants because they have minimal degradation, have minimal interference with biological processes among a number of other desirable properties. But even these materials (think Silicone for breast implants) still result in fibrous capsule formation.
Our immune response largely targets proteins and other organic compounds. This is for good reason -- generally the things trying to kill us are all viruses (coated in proteins) or bacteria (also coated in some proteins).
We don't really have much of an immune response to things like metals or non-reactive inorganic compounds. Our immune systems largely ignore these things.
Coating the chips properly with a suitable material will almost guarantee 0 immune response, basically.
There is ongoing research for better threads (sticking materials ) and coatings that are more bio compatible and don't get attacked by the immune system or are harmful to the cells. This is not done for micro chips, but for implants . Like for example hip or knee implants where the whole bone or parts of it needs to be replaced