Some 30 months since it was part of the accepted answer I'd like to disagree with
A dirty clay cup would not cause a disease either as that's not the route for transmission of most of those diseases
The good news:
It is not generally appreciated that just bringing water to about 90 degrees C will kill almost all pathogens instantly (as stated in the short but excellent Water treatmemnt page referred to by vistet.) The exception are nasties which form physically large cysts, such as Giardia. THese can be dealt with with quite simple filtration - but that's probably not useful for premade tea.
Based on my past investigations I'll state that all pathogens will be killed instantly if water is brought to even 90 degrees C,
close to instantly (say 10 seconds) at 80C
and for most cases, 'after a short while' at 70 C.
The exceptions in the last case MAY be large sized objects which form a cyst such as Giardia.
The bad news:
While the major route of transmission will be via contaminated water if it has not been purified, secondary recontamination from cups, hands and airborne material is a very well known means of introducing disease into previously sterilised liquids. I have carried out extensive investigation into methods for provision of safe drinking water and it is invariably necessary to take active steps to avoid post sterilisation contamination and/or to ensurte there is resdiual sterilisation component available in the liquid.
Residual sterilisation effect may be applicable with chlorinated water or when usimg eg filter pots using colloidal silver which leach small amounts of si;lver into the output water, but is very unlike;y to apply to drinks from street vendors.
One possible solution is to wipe or rinse the inside of the cup with a sterilising solution.
This is liable to at a minimum detract from the authenticity of the occasion and will likely attract some attention and make you look like a crazy westerner.
Possibly useful sterilising materials include
chlorinated water (could be stronger than usual with a small amount swirled around the cup and then discarded and the cup shaken dry,
alcohol gel (or straight alcohol),
or a suitably dilute benzalkonium chloride (BAQ) solution.
Alcohol that has been treated with a bittering agent will wreck a large amount of drink with a small amount of 'flavor'. Chlorine solution needs to be strong enough to be fast acting and weak enough to not be excessively 'tasty'. BAQ can be reasonably dilte. Some reading is in order, but probably 0.01% - 0.1% range may be OK. Stronger is liable to start tasting a little. Too weak and effectiveness rate will drop. Some systems use this at more like 0.001% - 0.01% range if exposure is long enough.