Historically, the spelling was "tire" and is of French origin, which comes from the word tirer, to pull. The reason for this naming is that originally "tire" referred to iron hoops or thick wires bound to carriage wheels. In French blacksmithing the word for a drawn iron rod is a tirer, or pull. The same word was often used for any metal drawing or rolling process. In an article in the London Magazine/Intelligencer of 1853 "The Utility of Broad Wheels," it explains that the common practice was to bend two rods, called "tires," into hoops and bind them to the wheel, but it is preferable to use an iron band, called a "broad wheel" rather than the rods, because as the rods wear they bite into the wheel. Another early mention of a tire in English is in The Scots Magazine, Volume 15 By James Boswell (1753).
Another origin of "tire" is provided by Online Etymology Dictionary, essentially that the word is a short form of "attire," and that a wheel with a tire is a dressed wheel. Some other etymologists may share this view.
The spelling tyre does not appear until the 1840s when the English began shrink fitting railway car wheels with malleable iron. Nevertheless, traditional publishers continued using tire. The Times newspaper in Britain was still using tire as late as 1905. The spelling tyre, however, began to be commonly used in the 19th century for pneumatic tires in the UK. The 1911 edition of the Encyclopædia Britannica states that "[t]he spelling 'tyre' is not now accepted by the best English authorities, and is unrecognized in the US", while Fowler's Modern English Usage of 1926 says that "there is nothing to be said for 'tyre', which is etymologically wrong, as well as needlessly divergent from our own [sc. British] older & the present American usage". However, over the course of the 20th century tyre became established as the standard British spelling.