Can an overdue oil change make a lawn mower burn oil?

Yes - over due oil change can accelerate wear and cause oil consumption.
No - changing the oil will not repair damage already done, but it will slow further wear.

Oil burning is a symptom of physical wear. Changing oil will not repair that. You might reduce the oil consumption by adding heavier oil or the suitable additive, but the real remedy is to recondition the engine.

Yes, not changing the oil can cause damage to the piston, cylinder wall and piston rings which will cause oil burning. Changing the oil will not fix the damage after it occurs.

It's possible, IF that oil change was several YEARS overdue. But typically, a small engine burning oil is due to worn piston rings.

Replace with SAE30 oil and breathe the smoke, or buy a new one for less than the rebuild cost.

This site can help
https://www.hunker.com/13404487/causes-of-a-lawnmower-burning-oil

Yes, an overdue oil change can cause oil consumption but changing the oil does not reverse the damage already done. Once the engine starts consuming oil, it will continue to do so until it is properly repaired.

Yep yep yep 200% change it before it blows up in your face and the lawn mower blade slices through your testicle!