Well, not to reduce the friction of the head against the tape of course
The head rather rides (or is supposed to at least) on two shafts (in my HP LTO-2 drive those are some white plastic material, in newer ones they are made out of metal) along the width of the tape, driven I believe by a single coil on a 3rd shaft.
When inserting a tape, the drive will move the head up and down once or twice (not sure, whether that is some kind of self-cleaning or self-calibration procedure). In the problematic drive it fails to do that, instead attempts some eight to ten times, thereby quite audibly knocking the head on the bottom, but not able to reach the full height. Turning the drive on the side, the knocking isn't quite so loud, but still the head moves only a fraction of the full width. Turning it up-side down yields similar results.
http://sbfmdrv.blogspot.com/2018/11/disassembling-and-cleaning-lto-tape.html describes the symptoms and cure and has some high resolution pictures.
I attempted the same cleaning procedure, found indeed some residue, but not a whole lot and after cleaning those the symptoms remained. I then (foolishly, I knew, but desperately
) cleaned the the shafts (and only those) with alcohol (70% isopropanol). In the beginning the head seemed to ride with more ease (when moved manually), but the drive still didn't succeed in moving it itself. Letting the alcohol evaporate over night, next day the symptoms remained and the head seems now to move even less freely. Not sure, whether the alcohol dissolved some not visible grease on the shafts or in the bearing of the rollers running on those.
Now the question for you fellows: what oil or grease would be used in such a situation? Is the shaft being greased or just the rollers?
(there's no rush really, as I just got on fleebay a *new* LTO-3 drive for a very reasonable price, which thankfully reads those old tapes of mine -- still, I'm curious)