Does the order of conditions in a WHERE clause affect MySQL performance?

The order of columns in your where clause shouldn't really matter, since MySQL will optimize the query before executing it. But I suggest you read the chapter on Optimization in the MySQL reference manual, to get a basic idea on how to analyze queries and tables, and optimize them if necessary. Personally though, I would always try to put indexed fields before non-indexed fields, and order them according to the number of rows that they should return (most restrictive conditions first, least restrictive last).

Here is a demo showing the order of WHERE clause conditions can make a difference due to short-circuiting. It runs the following queries:

-- query #1
SELECT myint FROM mytable WHERE myint >= 3 OR myslowfunction('query #1', myint) = 1;

-- query #2
SELECT myint FROM mytable WHERE myslowfunction('query #2', myint) = 1 OR myint >= 3;

The only difference between these is the order of operands in the OR condition.

myslowfunction deliberately sleeps for a second and has the side effect of adding an entry to a log table each time it is run. Here are the results of what is logged when running the two queries:

myslowfunction called for query #1 with value 1
myslowfunction called for query #1 with value 2
myslowfunction called for query #2 with value 1
myslowfunction called for query #2 with value 2
myslowfunction called for query #2 with value 3
myslowfunction called for query #2 with value 4

The above shows that a slow function is executed more times when it appears on the left side of an OR condition when the other operand isn't always true.

So IMO the answer to the question:

Does the order of conditions in a WHERE clause affect MySQL performance?

is "Sometimes it can do."

No, the order should not make a large difference. When finding which rows match the condition, the condition as a whole (all of the sub-conditions combined via boolean logic) is examined for each row.

Some intelligent DB engines will attempt to guess which parts of the condition can be evaluated faster (for instance, things that don't use built-in functions) and evaluate those first, and more complex (estimatedly) elements get evaluated later. This is something determined by the DB engine though, not the SQL.