The solid electrolyte in tantalum capacitors is traditionally manganese dioxide, actually a semi-conductor. It ain't going anywhere fast. Stored VERY damp, they will pick up moisture and might leak a bit - if in doubt put a very few volts across them and they will repair (NOT reform) - I would say measure the leakage current, but that is going to be so low...……….. unless you have a genuine dud.
Th other option is solid polymer electrolyte, although I'm unsure how common they are. There was a huge push to make them when I was working in tantalum R&D because they are potentially far cheaper to make. The same logic applies as with manganese dioxide.
They both have an almost indefinite shelf-life otherwise.
I suppose VERY long term they would go open circuit due to oxidation of the silver in the silver inks used as part of the connection between the cathode and the lead/lead-frame.