plugging tanner 1894 into search engine on manxnotebook (1894 = brown's directory - the last full directory) gives for tanners in Douglas
Killey Philip & Son, tanners, Market-street &
Litt Brothers, tanners and fellmongers, Market-street
(Market street (not much left now) has been much modified by modernisation of douglas - was orig Cattle Market st, according to Slack "streets of Douglas" the cattlemarket was on the site of Radcliffe's paper mill this being purchased by Philip Kelly to use as a tannery (it must have stank - I remember the smell of the old tannery which operated in Warrington upto late 1950's open baths of acetic acid etc) - (if you look at www.manxnotebook.com/fulltext/cg1843/map.htm then Market st runs to north of Sand street)
If I understand slack's description (if he had drawn a map his prose which relies on knowledge of now gone shops etc would be clear) it was somewhere near the chester st car park.

There was a tannery in aptly named Bark lane in Ramsey owned by John Lloyd