From Kevin Tisue,
Your Guide to Mountain Biking.
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You don't need to remove the whole tire from the rim. Simply unseat one side of the flat tire from the rim using tire levers if necessary.

When using tire levers insert the scooped end in-between the tire and rim and pry the tire out over the edge of the rim. Some levers have hooks on one end that grab a spoke to hold them in place while you work with another tire lever nearby. Try to keep the distance from tire lever to tire lever small or it will become difficult to pry the tire over the edge of the rim. Once the tire loosens up a little, you can zip a lever or your finger around to get the rest of it loose.