Indeed, research on this particular question is somewhat lacking, but what is available suggests that training does improve performance in tactile tests of the oblique effect.
In an article published 1999, Gentaz & Rossetti lament:
Unfortunately, the effect of practice on the haptic oblique effect has
not been yet studied.
An indication that this might soon be remedied was noted in Junker-Tschopp, Gentaz & Viviani (2010):
We used bars instead of fuzzy stimuli (such as Gabor’s patches)
because this study is part of a larger project in which we also tested
learning effects in the haptic modality (blindfolded participants
explored manually a rod with varying orientations).
It does not look like the results of this "larger project" were ever published. However, they are reviewed in the book Psychology of Touch and Blindness by Morton A. Heller and Edouard Gentaz (one of the authors of the above mentioned papers) published in 2013 (pp 56-57):
... Junker-Tschopp, Gentaz, and Viviani (2010) investigated the extent
to which visual and haptic perception are penetrable by cognitive
factors by assessing the effect of learning on the perception of
orientations. The "oblique effect" questions the nature of the
linkages between perception and cognition. ... In the haptic modality,
the results showed that practice resulted in a general dramatic
improvement of orientation discriminabilty. Furthermore, the
anisotropic bias in orientation perception is actually suppressed by a
global learning paradigm ...
These results are in line with the idea that the persistence of the oblique effect across modalities indicates that it is situated in higher cognitive processes (it is a top-down effect). This theory is backed by evidence demonstrating the similarity of the effect across visual and tactile modalities, so the effect of training would be expected to be similar in both.
Unfortunately, not everything is the same about the oblique effect across modalities, and authors note differences such as the role of gravity cues, memory constraints, reference frame, and while training does improve discrimination, it never eliminates the oblique effect, suggesting some bottom-up influence, and hence the potential for differences in results.
In a recent review, Mier (2014) examines the influence of training and feedback on the haptic perception of parallelity, and comes to a more guarded conclusion suggesting a more significant role for bottom-up processes:
... Kappers and coworkers (2008) found that the deviations were only
marginally affected by training and feedback. Without informing their
participants about their biased performance, they studied the effects
of visual training (seeing the correct orientations), haptic training
(feeling the correct orientations) and combined visuo-haptic training
(seeing and feeling the correct orientations). In addition they
studied the effect of error feedback on the performance of the
participants, again under visual, haptic and visual-haptic conditions.
They showed that the robustness of the deviations persisted even after
participants received haptic and/or visual feedback and training.
Haptic or visual training did not significantly decrease the magnitude
of the deviations, only combining both training modes resulted in a
small but significant improvement. ... Although providing participants
with haptic and visual feedback about their errors reduced the
deviations, performance was still far from being veridical ... As the
authors state, it might be that extension of the feedback phase would
eventually lead to more veridical parallel matching.
Compared to the visual realm, where training has failed to eliminate the oblique effect, the effectiveness of training on the tactile oblique effect is unclear. The first study suggests that training is more effective on the tactile oblique effect, and is capable of entirely eliminating it, while the second study suggests that the tactile effect is less amenable to training than in the visual modality. Note however, that the second study refers specifically to haptic perception of parallelity, which is a specific sub-case of the oblique effect where exceptionally large oblique effects have been found, and therefore may not be applicable in general. More research is needed.