Note: This answer uses my own examples, because I couldn't think of any examples in published work off the top of my head. Due to an ongoing bug, I couldn't post a comment asking if this format would work for you. So I went ahead and posted the answer.
The advice to use urgency with exposition is good advice, and I'll show you how below. It's obvious (and important) WHY this is good advice: exposition, whether done by the author or a character, pauses the story, and is inherently devoid of tension. It kills whatever action or intrigue you had going, and the reader immediately wants to get back to that.
So how do you magically incorporate this tension/urgency into exposition?
Obviously, a basic understanding of tension is needed. At it's core, tension is an unresolved question. Humans are by nature curious, and dangling a question in front of a reader automatically makes them want to know the answer.
Urgency is a special kind of tension. In urgency, the question is implied. It's not, 'will they succeed', but, 'will they succeed in time'. However, urgency is just one kind of tension, and certainly not the only kind of tension you can use with exposition.
A few examples
In the year 7334 DM, the Great Empire defeated the East, and enslaved them.
Many said the Eastern Rebels were a thing of the past, that the Great Empire had defeated them, but they didn't know the truth.
A slight difference. While I would still be leery about opening a book this way, it shows you what tension can do. Basically all of the relevant information is there, but one passage asks a question, while the other does not. The reader wants to know what 'the truth' is, and why the 'many' don't know it. And if this is the first line of a book, they obviously also want to know who the Empire and Rebels are, and what the war was about. But those are minor questions next to 'the truth'.
I asked a similar question to this, specifically about info dumps in fantasy novels. (Link: https://writing.codidact.com/questions/15692) If you look at Lostinfrance's answer, he has some good (if super short) examples. He has some excellent suggestions, which I'll sum up here:
- Character conflict - One or more of those participating in the conversation are at odds with each other, either visibly or internally. A great example is when the detective questions the suspect in almost every crime novel or show. That's pure exposition, but there's ample tension in almost every word (if it's done right). Sometimes the author mixes things up and has the suspect act contrary to what is expected, which is again just another form of tension (since it asks the question, 'why are they acting this way').
- Urgency - Understanding is vital before the time limit is up. Think of a bomb which has to be diffused. If you for some reason want to include exposition about how to diffuse a bomb, characters can talk through it, but the tension of the bomb ticking down, maybe hitting the wrong wire, uncertainty if it worked... that's all tension, and it keeps the reader asking himself the question: 'Will it work? Will they succeed?'
- Build up - Lostinfrance presents a great alternative to those methods. If you can't include or don't want tension during the exposition, build up the question before it, so that the exposition itself is the much anticipated answer, and the reader wants to read it. The method is simple. You know how to introduce tension: get the reader to ask himself a question. Make a statement that implies there is something the reader does not know, and he will want to find out that something. Well, to build up tension, just do that over and over. Be aware that the more hints you drop that there's something big the reader doesn't know, the more the reader will want to know it, and the less time you have to deliver. If you go several pages without delivering, the reader's interest starts to turn to frustration. While this method is great, you should keep the exposition (the actual answer) short. Reader satisfaction only carries so far.
While these are great, I understand your problem. So below, I'm going to write my own examples, using the previous three bullet points. Hopefully you can see how I keep the reader asking himself questions.
Danny stood, a fake smile stretching his face.
"Alright, dirtbag," he said, "here's the deal: either you cough up where your partner is going to be tonight, or you're going down for the murder of Jane Andrews."
Across the table, Kyle didn't react. He didn't lift his eyes, he didn't shift his slouched posture. But Danny saw the beads of sweat begin to accumulate on his bald head. He saw the slight twitch of the mouth. All it would take was a little extra coaxing.
"Okay," Danny said, sitting down and opening the folder in front of him. "I understand. Thieve's honor, right? You don't want to rat out your buddy. I get it."
He turned to Maria, his partner. "What's the minimum for first degree murder?" he asked casually.
"Thirty years, isn't it?" Maria supplied.
"Thirty years," echoed Danny, shaking his head. He made a note on the folder. "Without parole, too." He sighed.
"Alright!" Kyle said, finally breaking. He was sweating for real now, a look of desperation in his eyes.
Danny smiled. "Where's your partner, Kyle?"
"North bank. Intersection of Second and Beecher. But you'll never catch him. He can smell cops coming."
"And why's that?" asked Maria.
Kyle shook his head, swallowing. "A rat," he said. "He's got a rat, here in the precinct. He'll know the second you leave."
"Look Mike, either you tell me how to diffuse this thing or the radius is going to take you out too."
"Okay, okay, give me a second. What's it look like?"
"Wire, circuit boards... it looks like a bomb, Mike! What did you think it looked like?!?"
"Okay, calm down. Look for a trigger. Do you see a phone, or any kind of antenna?"
"Yeah, there's a phone. It's blank."
"Okay, you can't just rip the phone out. How many wires are coming off of the phone?"
"Two. One goes to this round cap, and the other hooks up to a motherboard-like thingy."
"Good. Cut the one connecting to the cylinder. Be careful - that's the explosive. The other wire is an anti-tamper lock. If you move the phone, that wire will send a signal to the computer and the whole thing will detonate."
I suppose there had been signs. Plenty, actually. I mean, come on. Who else wakes up with a five hour blank spot in their memory? Someone who had drunk too much at the party? Sure. But with blood on their hands? Shards of glass in their boots? The smell of gunpowder on their clothes? No, there was only one explanation, and I didn't like it one bit:
I was the shooter.
Hopefully these examples give you a good idea of where to go. Notice how each provides essential details, but they all do so in an atmosphere of tension.