In many places in Northern England, "burnie bee" refers to a ladybug.

"The lines are said by children when they throw the insect (a ladybug) into the air to make it take flight." -Mother Goose's Melodies for Children

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The version from "Harry's Ladder to Learning" (1850)...

Bless you, bless you, bonnie bee:
Say, when will your wedding be?
If it be to-morrow day,
Take your wings and fly away.

Here's some information about how "burnie bee" is a ladybug (also called lady-bird) in northern England from Mother Goose's Melodies for Children, Or Songs for the Nursery With Notes, Music, and an Account of the Goose Or Vergoose Family by Henry Louis Stephens, Gaston Fay (1869):

"Burnie bee is the name given in the north of England to the ladybird. The lines are said by children when they throw the insect into the air to make it take flight. Variations of the song are current throughout the north of Europe."

We found the following about Burnie-bee from Notes and Queries (January 25, 1902)…

"In Norfolk the May bird is called burnie-bee, by contraction from burnie-beetle or fiery-beetle. The following address to that insect is in the mouths of children there :-

Bless you, bless you, burnie-bee,
Tell me where my true-love be ;
Be she east, or be she west,
Seek the path she loveth best :
Go and whisper in her ear
That I ever think of her ;
Tell her all I have to say
Is about our wedding-day.
Burnie-bee no longer stay ;
Take your wings and fly away."

Notes and Queries got the rhyme from The Monthly Magazine or British Register (1814). The 2nd illustration is of this version of the rhyme.

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James Orchard Halliwell wrote in "Popular Rhymes and Nursery Tales" (1849):

"In Norfolk the lady-bird is called burny-bee, and the following lines are current:

Burnie bee, burnie bee,
Tell me when your wedding be.
If it be to-morrow day,
Take your wings and fly away."

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Thanks and Acknowledgements

This rhyme can be found in The Real Mother Goose (1916), illustrated by Blanche Fisher Wright.