They told me it draws only a teeny tiny amount of current, on the order of tens of microamps (that's way way less than 1 milliamp)
If it's 30 microamps that's 0.03 milliamps. My unit uses 4 AA batteries, each of which should be around 24,000 mAH. So they should last 80,000 hours, which is about 9 years.
That figure of 0.03 was just a guess and could be off by a factor of several. The batteries themselves will have a shelf life even with no drain. There are probably non-linear effects involved too.
I would refrain from removing and replacing the lithium batteries every time you use the unit
I agree, but I do think it's worth removing them for long-term storage.
(I've cut a plastic insert to break the connection, so I can keep the batteries inside the unit for their full shelf-life.)