Origins
The word 'tax' descends from Latin 'taxāre' (to evaluate, to estimate, to touch sharply, to censure), which is an intensive or frequentative form of 'tangere' (to touch), from PIE *tag- (to touch, to handle). The semantic development from 'touch' to 'tax' follows a logical chain: touching led to handling, handling led to assessing (evaluating something by handling it), and assessing led to imposing a levy based on assessed value.
The word entered English through Old French 'taxer' in the fourteenth century. The primary meaning was 'to assess the value of property or income for the purpose of levying a charge.' The noun 'tax' (the charge itself) followed from the verb. Over time, the word acquired the additional meaning of 'making heavy demands on' — 'this work taxes my patience' — which preserves the older sense of pressure and burden.
The semantic connection between 'tax' and 'taste' (both from 'taxāre') illustrates how a single root can diverge dramatically. 'Taste' followed the 'touching' sense of 'taxāre' through Vulgar Latin into a word about sensory perception. 'Tax' followed the 'assessing' sense into a word about governmental finance. Both are descendants of the same Latin verb, yet no English speaker would spontaneously connect them.
Latin Roots
The word 'taxi' (short for 'taxicab,' short for 'taximeter cabriolet') also descends from this root. A taximeter is a device that calculates the fare (the 'tax' or rate) for a journey. The French word 'taximètre' combined 'taxe' (a rate, from Latin 'taxāre') with Greek 'metron' (measure). Thus the taxi that drives you to the airport and the tax you pay on your income share an ancestor in the Latin verb for touching and assessing.
'No taxation without representation' — the rallying cry of the American Revolution — uses the word in its core financial sense. The complaint was not about the touching or assessing but about the imposing: the colonists objected to being taxed (assessed and charged) by a Parliament in which they had no voice.