The phrase never put anything smaller than your elbow in your ear was explained as follows in Guidelines: Put nothing smaller than your elbow in your ear, published by Industrial Safety & Hygiene News (Birmingham, Michigan, USA) on 10^th January 2017:

Updated clinical guidelines published the journal Otolaryngology-Head and Neck Surgery say cotton swabs are not appropriate for earwax removal. In fact, information for patients in the guidelines say not to put anything “smaller than your elbow in your ear.”
[…]
Here’s why not: Cotton swabs, hair pins, house keys and toothpicks—the many smaller-than-our-elbow-objects we love to put in our ears—can cause cuts in our ear canals, perforate our eardrums and dislocate our hearing bones. And any of these things could lead to hearing loss, dizziness, ringing or other symptoms of ear injury.

The earliest occurrences that I have found of the phrase never put anything smaller than your elbow in your ear, and of its variants, are as follows, in chronological order:

1-: From one of the unconnected paragraphs making up column 4, page 3, of the Grand Island Times (Grand Island, Nebraska, USA) of Thursday 14^th August 1879:

Don’t fool with your ear. Never put anything smaller than your elbow in it.

2-: From one of the unconnected paragraphs making up the column American and European Brevities, published in the Maryborough Chronicle, Wide Bay and Burnett Advertiser (Maryborough, Queensland, Australia) of Saturday 12^th March 1881 [page 4, column 1]:

A Dimondale woman lost a pin in her ear, and is having agony by wholesale. Moral: Never put anything smaller than your elbow in your ear.

3-: From Look Out for the Ears, published in the Huron Times (Sand Beach, Michigan, USA) of Thursday 21^st June 1883 [page 6, column 2]—reprinted from the Toledo Blade:

It is […] a common custom to drop all sorts of hot things into the ear or to syringe it with warm water. Never do either unless under the direction of a physician, for while you may sometimes apparently do good, more often the final effect is irremediable harm. The very construction of the ear, the situation around the outer part of the auditory canal of the glands to secrete the bitter wax, the purpose of which is evidently for protection against the incursion of insects and to prevent the growth of mould, shows that Nature did not intend foreign substances to be introduced unless with the greatest care.
[…]
An old physician once quaintly said: “Never put anything in your ear smaller than your elbow.” Of course he meant to give the same advice that I have given, and which the best aurists of to-day will sustain me in.

4-: From Our Babies and their Mothers. Claims of the One and Duties of the Other. Chapter II. Instruction for Bathing and Feeding, by A. A. Whitfield, M.D., published in Good Housekeeping. A Fortnightly Journal. Conducted in the Interests of the Higher Life of the Household (Springfield, Massachusetts, USA) of Saturday 26^th May 1888 [page 32, column 2]:

I cannot leave the subject of baby’s toilet without mentioning the insane impulse some have of shoving hairpins, shawl-pins and various other articles up the nostrils and into the ears of the poor victim, for sundry and various purposes as a fitting finale of the cleaning up process. I have as strong an aversion to dirty ears and nose as any one, but don’t jeopardize your child’s life by using such tools. You are apt to puncture delicate organs which lie in these cavities. It’s a good saying: “Never put anything smaller than your elbow in your ear,” but when necessary to explore these cavities, use only a corner of the towel or something not too hard.