Common idea: INT heroes are good early-mid game but they become quite useless late game. Wrong. Actually, INT Heroes are strong early-late. It's just their usage and playstyle that changes. In the laning phase, they're important because of the sheer power of their nukes and stuns which is extremely useful for any team situation. Not to mention their ability to arrive on the scene and kill quickly (which gives flexibility for planning strategies spontaneously.)

In fact, I believe that INT heroes are 'carry carries', because they're the ones that hold the game long enough for the true carries to get powerful. Which is why they're picked in each and every game.

So let's get to the topic: How INT heroes can be useful late-game (even in public games)
For one thing, we know that some of the INT heroes can become semi-carries (sometimes, the main carry) late game. Examples are Nortrom, Obsidian Destroyer, QoP, Alleria, so on.

Though some may disagree, nukes are still powerful late-game because of their RELIABILITY. Look at my 'Understanding hero combos' post for more details. Nothing beats arriving to the scene and dealing quick and instant burst damage. I'd like to call nukes as the early game 'EARLY GAME OBLITERATORS' and the 'LATE-GAME CARRY COUNTERS'
Early game, you dominate by relying on your nukes' damage. But how they're useful late-game is what's interesting.

Late game is when most carry heroes rely on their physical attacks (Troll, Morphling, etc.) and have to stand and trade hits to deal damage. So when you're up against such a hero (farmed), standing and trading hits with him is not the best idea. A better idea is to stall for time and waiting for the initiators/nukers arrive. Nukers don't have to trade hits; because they can quickly and efficiently deal damage regardless of how strong the opponent is. This is what makes them important 'carry counters'. When it's an AGI vs. AGI it depends on who has more DPS (bash chance, lifesteal, etc. also comes into play; but let's keep things simple). The weaker person can't hold his own in a stand-ground duel vs. the enemy carry. But NUKERS can ninja a spell from afar.

Let me illustrate this with an example: Zeus, an INT nuker.
Early game your nukes hurt hard, and if you play your cards right, the game shouldn't last very long. Mid-game you take on the role of a ganker. Late-game (where most people believe you're weak) is where it's different. Now, heroes can take on your spells and still survive. So you have to move into the 'second-line' of attack and change your playstyle to 'ninja' mode. Before the bulk of the battle begins, try to get in a Lightning Strike on a key hero; it helps more than you think it does. Even a -200 HP to start with makes your team target that hero first. Also you're good at initiating a 'focus-fire sequence' because of the huge casting spell effect. Improvise.

There are loads of other things which you can do late-game as well. Here are a few:
Initiate 'focus-fire sequence' with a Guinsoo (which most people buy for INT heroes); provide solid support to your team in the form of a Shiva's Guard; massive AoE, mind games (because of your low cooldown-spells and a disable if you've purchased a Guinsoo), ability to counter carries, so on and so forth. The possibilities are endless.

Sometimes, overfarmed carries tend to think gung-ho and look down on their team's casters and nukers. And you know what'd happen to such a team.