Just as a quick answer, because apparently some people are in doubt so let me try to clean up the confusion.
The 'physics part'
Surface tension that people normally talk about is the liquid-air surface tension, in this case water-air, and I am pretty sure this is the surface tension you are referring to in your question. This water-air surface tension is a property of the water and air and has absolutely nothing to do with the physical shape or material of the container.
For completeness, I have to mention that the contact with the walls that have different chemical properties will change the shape of the water-air interface a bit, which you might misinterpret as a change in water-air surface tension but this is caused by the water-container and container-air surface tensions which indeed DO depend on the container.
The 'chemistry part'
What could * change the water-air surface tension is the exchange of some material out of the container into the water or the other way around. This would mean that you are not really looking at the water-air surface tension anymore, but rather at the (water$+$some solute)-air surface tension. Which might indeed be different, typically lower.
If this exchange is the mechanism then the time over which the surface tension will change is dependent on the kinetics of that process and without knowing what is being exchanged it is impossible to tell you what the time span will be, it could be seconds, it could be hours. Note that the explanation with stuff leaking out of or into the lota would mean that (assuming there will be equilibrium at some point) storing water in the lota and then drinking it from the glass would be the same as drinking it directly from the lota.
*I'm not saying it does I have no clue whether this happens with a lota