In British English, these are the spellings for two words with different meanings.

Tire:

• to reduce the energy of (usually by exertion); to feel or cause to feel in need of sleep/rest
• to become bored with or lose interest in something; to exhaust the patience of.

Tyre:

• a ring of rubber fitted round the rim of a wheel of a road vehicle to form a soft contact with the road
• a band of metal fitted round the rim of a wheel (typically of a railway vehicle).

In American English, tire is the standard spelling for both meanings. Tyre is actually thought to be a variant of the old form tire (a shortening of attire because the tyre was seen as the ‘clothing’ of the wheel).

Thank you to Aeryn Rudel for suggesting this post.