The implication of "walk it off" is not just ignore the problem and hope it gets better, but instead that the walking will actually be beneficial.

For example, in baseball, if someone is hit in the leg by a pitch is is much more reasonable to say "walk it off" than if they are hit in the arm or head.

For an older example, there is the 1736 play Eurydice Hissed, or a Word to the Wise:

[Mr. Emphasis]: Why, faith, Jack, our Beer and Beer sat but ill on my Stomach so I got up to try to see if I could not walk it off.

 

[unnamed character]: I wish I had any thing in my stomach to walk off...

The implication of "walk it off" is not just ignore the problem and hope it gets better, but instead that the walking will actually be beneficial.

For example, in baseball, if someone is hit in the leg by a pitch is is much more reasonable to say "walk it off" than if they are hit in the arm or head.

For an older example, there is the 1736 play Eurydice Hissed, or a Word to the Wise:

[Mr. Emphasis]: Why, faith, Jack, our Beer and Beer sat but ill on my Stomach so I got up to try to see if I could not walk it off.

[unnamed character]: I wish I had any thing in my stomach to walk off...