stales, it may be that we have two phrases (i.e., give it a burl/whirl) linked by coincidence (or not). the folks at OED don't seem to have made a direct connection, for they give us:

burl n. [App. var. north. dial. birl a whirring sound, a rapid twist or turn, etc. (see E.D.D. and Sc. Nat. Dict.); cf. birl v. to revolve, rotate.] 1. ? Roundness, fullness. poet.
2. Also birl. An attempt, a try, a test; esp. in phr. to give it a burl, to make an attempt. Austral. and N.Z. colloq.

birl v. Scot. [A modern word: apparently onomatopœic; having probably association with birr, whirr, whirl, hurl, and Sc. dirl, pirl, in all of which there is a reference to vibration or rotation and its sound.]
1. intr. To move on with rotatory motion, as a rifle bullet; also fig.
2. To revolve or rotate rapidly and with characteristic noise.

the vbl n. birling would seem to fall out of the lattermost sense.

-joe (I'm a lumberjack and I'm okay) friday