I read somewhere, forget the source, that one should apply a dab of lubricant such as cork grease, to a whistle's tuning slide once a month or so. Is this a common practice?

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A moment of carelessness, a lifetime of regret.
A lifetime of carelessness, a moment of regret.

The first "source" I'd pay attention to, is the manufacturer. There are a variety of materials and designs used for the slide section of a whistle, and different materials require different types of maintenance. O-rings, synthetic wrap, brass slide, cork, etc., they should all only get exactly what the manufacturer recommends.

If you can't get a response out of a manufacturer for any reason, determine what the slide material is made of and if you can find another model of whistle with the same material being used, contact that manufacturer.

Slides….

Metal against metal: Petroleum-based grease
Metal against o-ring: Silicone grease
Metal against plastic: Silicone grease
Plastic against plastic: Silicone grease, or a smear of beeswax

Most of my Low Whistles have a metal tenon and metal socket but there's an intentional gap there for Teflon plumber's tape, or waxed thread if you prefer. These require no maintenance.

I very much prefer this sort of tuning slide over the type with rubber O-rings like Burke and Susato.

I find that I have to take apart and lubricate Burkes on a fairly regular basis or they'll seize up.

I'm spoiled a bit by the tenon-and-socket joints on bagpipes, which have waxed "hemp" wrapped around the tenons. If you do a good hemping job the joint will work fine for years with no maintenance.

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Richard Cook
1978 Quinn uilleann pipes
1945 Starck Highland pipes
Goldie Low D whistle

Try Sil-Glyde on the Burkes https://tinyurl.com/yaysoacb