What is an instant of time?
I believe you're asking about a paradox in the style of Zeno's paradoxes. Your paradox is most similar to 'Paradox of the Grain of Millet'. You want to know how an infinite sum of infinitesimal instants could equal a finite length of time, $$ \int~\mathrm dt = t. $$ Well, the above is nothing but $$ \int~\mathrm dt =\lim_{N\to\infty} \sum_{n~=~1}^N t/N = t, $$ This, and many of Zeno's paradoxes, are resolved by understanding calculus and infinite sums.
If we say that an instant of time has no duration,
It does indeed have no duration, by definition. The word "instant" has no special physical meaning here. It means the same as labeling regular X/Y coordinates on a sheet of paper with a number.
Such a coordinate (be it in space or in time) is something fundamentally different than an interval between two such coordinates: the coordinate does not have a length, while an interval does have a length. The coordinate does not even have length 0, it has no length defined.
why does a sum of instants add up to something that has a duration?
This is where you are mislead. "Instants" (i.e., coordinates) are not summed up. Intervals are, by summing up their lengths. But the operation of "summing up instants (coordinates)" is not a useful operation here because instants (coordinates) do not have a length, so there is nothing to sum up.
N.B.: if you want to find out about the "real" mathematics behind this, check out Rieman Integrals, measure theory, "countably infinite", "almost everywhere" and such terms. There is a fascinating world behind what you discovered.
N.B.: as mentioned in the comments, you can of course indeed sum up "instants" (i.e., intervals of length 0), but only countably many, and a sum of countably many 0 is still 0. In the lazy formulation in my last paragraph, this is taken into account.
Perhaps it's useful here to differentiate between a specific time (as in, a one dimensional representation of a specific instant or location in time) and a duration, which is the measure of difference between two specific times.
In this case hat you refer to as a summable 'instant' may actually refer to a delta of duration, for example the Planck Time - named after physicist Max Plank, which is the amount of time it takes a photon to travel the Planck Length, which according to physlink.com is
roughly equal to $1.6 × 10^{-35}m$ or about $10^{-20}$ times the size of a proton
Which makes the duration of Planck Time equal to
roughly $10^{-44}$ seconds
I offer this explanation merely as an interesting aside to the already accepted answer - which I think probably better addresses your question.