Do research-only postdocs harm chances for tenure-track opportunities? Suppose a person spends their PhD at an institute with a high ratio of PhD students to other students, resulting in a very low teaching load for PhD students, resulting in no or almost no teaching/tutoring.  Suppose he/she spends years in postdocs, where each postdoc is funded by supervisor grants, and involves no teaching.  She/he does great science but does not acquire other skills or experience.

Consider that tenure-track positions always require stellar experience in research, teaching and getting proposals funded — all of the above.

Is it harmful to an academic career if research is great, but all done on other people's grants and with no teaching experience to speak of?  If yes, how could one prevent or overcome this?
It depends. At my R1 university where the only thing that counts is research, you would be an ideal candidate for our hiring committees. 

At other R1s which balance research against a modicum of teaching, you might face a little bit of an uphill struggle. It might be good if you could at least pick up some teaching experience or certification1 to help you in the job market.

You face the most problems at small liberal arts colleges, where you would face two issues:


The concern that you don't know how (or really care) to teach 
The concern that you would think yourself ultimately destined for R1s and thus leave the SLAC at the first opportunity possible


You would have to counter these two prejudices in your job letter and interviews.


1. Even at my august institution, we're realizing our graduates are having problems on the job market because they have little to no teaching experience. So we have instituted a teaching and learning center where they can take seminars and workshops on pedagogic methods. They get a little annotation (certification) to this effect on their transcripts. Certainly not comparable to the teaching experience that doctoral students receive at public R1s.