The words dire, dyer sound the same but have different meanings and spellings. Why do dire, dyer sound the same even though they are completely different words?
The answer is simple: dire, dyer are homophones of the English language.
Warning of or having dreadful or terrible consequences; calamitous: a dire economic forecast; dire threats.
Urgent; desperate: in dire need; dire poverty.
One whose occupation is to dye cloth and the like.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").