Term for dirt / grime from frequent touching

How about "It had developed a patina of grime from being frequently touched"?

https://www.thefreedictionary.com/patina [3]

(The reference cites "neglect" as the source of the "patina of grime," but I prefer adding "from being frequently touched" to invoke the phenomenon you intend.)

You might consider:

• An incrustation - defined as "a layer of material, such as dirt or a chemical, that forms on something, especially slowly".

• A buildup of dirt/grime. This is somewhat similar to saying an accumulation but is idiomatic in this context. The definition of "buildup" is "a gradual accumulation or increase, typically of something negative that leads to a problem".

• Ingrained dirt/grime. This suggests that the dirt is not just surface dirt, but has become deeply embedded through prolonged or repeated touching. One dictionary gives this example along with the definition: "the ingrained dirt on the flaking paintwork".

How about contact grime? From Lexico:

contact: The state or condition of physical touching.

grime: Dirt ingrained on the surface of something.

The OP "is looking for a concise way to describe the grime that accumulates on frequently-touched objects and surfaces such as doorknobs and handrails." Contact grime does the job as it describes dirt ingrained on the surface of something, e.g., door knobs or handrails, by virtue of physical touching, e.g., by the fingers or hands. The OP also said in a comment that he prefer something that strongly implies touching -- contact does that. Contact grime also implies frequent touching as it takes time for grime to accumulate.

This might be a controversial one because the the origin of the word is Yiddish, but the first thing that came to mind was schmutz.

https://www.lexico.com/en/definition/schmutz

schmutz (also shmutz)

Pronunciation /SHmo͝ots/ /ʃmʊts/ 
noun
informal North American

Dirt or a similar unpleasant substance

There was a popular TV campaign in New York back in the early 2000's for a newspaper (remember those?) that was printed with ink that did not rub off on your hands. The tagline was "No more Schmutz!". The term might not be universally known, but it definitely gained some popularity after that campaign.