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I clean mine every so often. The digital probes and the analog dial type as well.
It's also a good idea to test their accuracy every so often as well (ice water bath and boiling water check). Some digital models have a recalibration function, but how you do that varies by brand/model so check your manual. Usually a commercial analog dial thermometer can be recalibrated but most consumer grade ones cannot.
Meat probes must be cleaned before *EVERY* use for food safety reasons.
some Styrofoam for dividers (with the shapes cut into them) and all my Mavericks and Thermopens will be very safe.
Even if you put your smoker up for the Winter , you could keep the Analog therms. from your Smoker there too.
And cleaning often keeps them in good operating condition. The meat Therms.- you should keep some Bleach Water around to rinse them in to keep down cross contamination.Gloves too!