Phrases like "instant of time" are tricky. If you're not careful, you can dive down rabbit holes that lead to pedantic questions like "what does 'is' mean?" which do little but frustrate people. You've got a good question, just be prepared for the answers to be a little more dodgey than you might like.
One of the major keys to the question is in AnoE's answer. AnoE draws a distinction between an instant of time and an interval of time. This distinction is very useful because intervals of time tend to be pretty well behaved in our minds. We're really comfortable that 2 intervals of 1 second, when put end-to-end is an interval of 2 seconds. We can then add another interval of 1 second to our 2 second interval to get 3 seconds. 3 and 1 can make 4, and so on. Nothing suspicious here. We can even postulate that this process can go on "forever," even if we don't have a very strong definition of "forever" to work with. None of our concepts have ever been proven out as they stretch to "forever" because nobody has lived that long!
What about the other direction? If I have a 2 second interval, I can subdivide it into a pair of 1 second intervals. I can take one of those and break it up into a 0.5 second interval. That can be broken into 0.25 second intervals and so forth. It seems like you can repeat this process, just like we could repeat the addition of intervals, until we get to arbitrarily small intervals.
But what happens if we go further? What happens if we keep subdividing until the intervals are "infinitely small." As it turns out, I have to put "infinitely small" in scare quotes because that concept turns out to be really tricky to assign an exact mathematical meaning. Zeno's paradox was mentioned by innisfree. It's the idea that, if I were to try to walk from one point to another, I would have to travel half of that distance first, then I could try to walk to the endpoint. However, once I get to the endpoint, I also have to walk half way between that half-way point and the end point (a three-quarters point). You can repeat this process to show that it would take me an infinite number of steps to reach you. Zeno argued that that means you can never reach anywhere!
Of course, anyone over the age of 3 will point out that Zeno is wrong. We go places. It happens (or at least appears to happen, if you're a hyper skeptical individual). So there must be something wrong with the argument. The truth of the matter is that Zeno was not actually suggesting that we are incapable of moving, he was pointing out a fundamental issue with the way we conceptualize the world around us. He was pointing out that our models suggest the world acts in a way which it demonstrably does not.
You see lots of mathematical answers here because mathematicians were really the only one that grappled with infinity in a rigorous enough way to actually have to worry about Zeno's paradox. Most people could just say "oh, it works because... we see that it works every day." Mathematics and reality aren't divorced, but sometimes it sure looks like they're in a trial separation! They call things like Zeno's paradox "supertasks." These are tasks that take an infinite number of steps to complete.
And the mathematicians went forth and analyzed these supertasks. Over hundreds of years, they came up with incredibly powerful tools. For this particular topic, Calculus is the crowning achievement of their efforts. Calculus is a very rigorous way to handle these infinite numbers. More importantly, it is an example of where mathematics and reality agree -- the hypotheses put forth using calculus have an excellent track record of being provable in real life!"
And so we have tools to formally capture these concepts of an "instant of time." We can even aggregate them together into intervals using a calculus tool called "integrals." However, when we do so, we'll see an equation like $\int f(t)dt$, where f(t) is the value of f at "an instant in time." However, we can't ignore the "dt" part of that integral. It's the notation we keep around to remind ourselves of how calculus is actually handling intervals. You can get away with thinking of the "dt" as an "infinitely short interval," as long as you only do so formally. It has a very particular formal meaning that's slightly different. However, the key to all of this is that your intuition lines up with what the mathematics actually do. When you walk somewhere, you actually get there!
So it will be very hard to find a non-mathematical example of summing instants of time into something that has a duration, because it's very hard to capture the concept of "an instant in time" with enough formality to talk about summing them together outside of mathematics. I know of no other field which has a level of rigor sufficient to truly explore this concept. Philosophy might, though their approach may be very different.
I recommend two vsauce videos if you want to learn more. He's my personal favorite way to learn about mathematics, and especially tricky concepts like infinities and infitessimals.
- How to count past infinity - In this video, VSauce provides an excellent introduction to infinity, and more importantly, a great discussion of all of the quirky oddities that arise when you try to capture the concept of "infinity." I find it very helpful to see just how many contortions mathematicians must go through to capture these concepts of infinities and infinitesimals without generating really boggling results.
- Supertasks - This episode is direct to your question. It explores the concept of tasks that have an infinite number of steps yet complete in finite time
- The Banach-Tarski Paradox - If you feel like going down a rabbit hole, the Banach-Tarski paradox is an example of just how strange the world of mathematics can get as you try to apply "obvious" claims to infinite sets. It's not required reading, but if you want to see the sorts of fits people have trying to make sense of infinities, this is about as "special" as it gets.