Plated through holes for screws? Or plated through holes as attachment points?

If I understand your issue, you use the annular ring around the plated though hole as a test point (which is not always a good idea in the first place). You are now observing tarnishing (note: tarnishing is not oxidation) of the annular ring and other test points. It sounds like the tarnishing is being observed after surface mount reflow.

One option to prevent tarnishing is to design the stencil so that every surface with immersion silver is coated with solder paste and therefore the silver is consumed during the reflow process. Please note that this will not consume the immersion silver down the plated through hole. And some manufacturers have reported field issues with silver corrosion products migrating out of the plated through hole. These migration mechanisms tend to only occur in areas with high levels of sulfur containing gases (off-ramps, rubber factories, sewage treatment, etc.).

Besides visually observing tarnishing, is there a failure associated with the tarnishing?

Silver can indeed tarnish when exposed to different atmospheric contaminates. The most common forms are sulfides which readily form tarnish on the silver. If subsequent exposure to the end use environment or conditions within the factory interfere with the test points and other locations then certainly protection of these locations by printing and reflowing solder on them is a good idea.

Solder is more resistant than the silver to the same atmospheric contaminates. When coating test points with solder, if using no clean solder you need to be sure that the residues do not interfere with the ICT probing process. Choose a no clean paste that has been shown to be compatible with you probe/pin tester.