Fortunately, you can drink potions in first person, but with third person and backside view excluded, unfortunately...
MCPE-11786
You cannot drink any potions in third person, and backside view
Resolved
Invalid
0
(Unassigned)
Confirmed
Tablet - Android - Samsung Galaxy Tab 2
Android 5.1.1
0.13.0 Beta 1, 0.13.0 Beta 2, 0.13.0 Beta 3, 0.13.0 Beta 4, 0.13.0 Beta 5, 0.13.0, 0.14.0
Retrieved
You can drink potions, it is just missing animation in the mentioned views.
Reported for fix.