Not much without permanent alterations.  A leather strip will help.  A thin bent metal or hard plastic plate tucked under the strings would last longer. 

Most of us just sort of accept that the round-over of the fretboard is going to notch if the builder did not install a piece of fretwire (like your aluminum rod) at the break-over.  Your aluminum rod will notch under the pressure of steel strings.

Generally the cutting-in goes 'so far and no farther'.  I have dulcimers that are decades old, and the cutting-in has not continued much more than the depth of the string diameter.  

New dulcimers going out of tune after a day or so is quite common, and not just caused by the string cutting it.  New strings take a day or two to stretch into the root tuning that you use.

updated by @ken-hulme: 10/14/18 09:39:21PM

Hello all,

I picked up my first dulcimer several months back and found that it would go out of tune after a day or so. After further investigation, I noticed that the middle and melody strings had cut in to the tail and were going deeper with each tuning.

Since I hadn't paid much for the instrument, I decided to solve the problem by cutting the tail at the top end and placing a bit of aluminum rod for the strings to rest on... This dulcimer hasn't gone out of tune since.

I have since acquired a number of dulcimers and am having the same problem with two others but don't want to modify them, as they are much nicer and were quite a bit more expensive.

Does anyone have any suggestions on how I can stop this without permanent alterations?