Here are some examples of sentences using each word:
Die:
* The old tree died after a long, hard winter.
* The soldiers died bravely in battle.
* I died laughing at your joke.
Dye:
* She decided to dye her hair a vibrant shade of red.
* The fabric was dyed with natural indigo.
* They used a special solution to dye the wood.
The key is to understand the context and use the correct spelling based on the meaning you intend to convey.