Yes there are some issues with Hand Sanitizers.

Hand Sanitizers are made with Glycerin. Glycerin is considered a "contaminate" for circuit boards. It can negatively affect circuit boards because it is Hygroscopic and can attract moisture to the Circuit.

A lot of this used to be used in HASL Fluxes and is a no go for Circuits that are not going to be washed. They too could dissolve the natural Halides from your fingers and transfer the Halides to the Circuit with added moisture leads to potential failure.

Alcohol Based Sanitizers are also like alcohol based napalm, it is highly flammable, and when lit it considered to be a white flame (hottest that a flame can be).

There are Antibacterial Lotions is being used by electronic assembly plants that do not contain any of the contaminants of PCB's Ex. Lanolin, Glycerin, Mineral Oils, or Silicones.

You should probably have your company provide these approved lotions on the factory floor and restrict the use of personal lotions.

Absolutely you should be concerned. Oils, both natural and those in various sanitizers can easily cause surface contamination on sensitive/delicate surfaces.

If you are using original scent hand sanitizers which are mostly alcohol I would think that it would be OK.

Just stay away from the scented or "enhanced with moisturizers" as they may carry contaminates to your process.

If they were not using hand sanitizers would they be handling boards with bare hands? Any contact with a bare hand is going to leave Sodium and Chlorine salts on the surface of the board, from finger prints.

I think its just good practice not to handle boards with bare hands. It some what depends on your industry. I have always worked with products where high reliablity was expected.