In the first Modern Warfare, Outlaw sees an escorting Cobra get shot down and, despite the nuclear threat, they decide to try and rescue the pilot of the Cobra. It's pointless as the nuke goes off just seconds before Outlaw reaches minimum safe distance.
The impact of the scene is enhanced by the fact that Price reflects on the long history between him and Soap as various pictures of Soap and Price are shown in cheery moods during their Task Force 141 days (Ghost appears in one of these pictures too along with Sandman). Price then says regarding Soap that you don't really appreciate your friends and their accomplishments until they are gone and you try your best to keep the fond memories of them alive even as their faces fade from your memory.
In Modern Warfare 3, shortly after Soap dies, Baseplate aka MacMillan consoles Price upon hearing of what happened, even calling him "Son" and in his voice you can tell he really gives a damn about his protege, you get that fatherly vibe off him.
The fatherly vibe is present in every conversation. Note that he is the only person in any of the games to address Price by his first name, John. Even as late as the current game, the Master-Apprentice Chain remains intact.
A small but significant Heartwarming moment appears in the briefing of "Dust to Dust" - when the logo of Task Force 141 is shown, the "DISAVOWED" ribbon is no longer blemishing the aforementioned seal of true badassery. Price did it. He won. He cleared their names of treason and acts of terror. Now if only Soap and the rest of Task Force 141 had lived to see it...
The fact that this is shown immediately after Price's briefing wall shrine drives it home.
There's a subtle bit of symbolism in the end where Price strikes his lighter once Makarov is dead: It takes three strikes to get it to finally light. One for Soap. One for all the other comrades he's lost, like Sandman. And finally, the last light for himself. One can almost imagine the ghosts of his friends sitting next to the old soldier and enjoying the well-deserved victory cigar.
A sort of implied heartwarming moment happens throughout 3. By this point, Price is considered a terrorist and traitor, and is officially wanted by the US Government. Sandman however works with him and takes intel from him, because he implicitly trusts Price. He knows and trusts Price enough to know that the charges against him are false, and that Price is not a terrorist. The rest of Team Metal also work with Price in "Down the Rabbit Hole", so either they also know and trust Price enough to do this; or Sandman simply told them Price can be trusted, and their faith in Sandman is strong enough to take him at his word.
A bit of Truth in Television here, as the special forces community even at an international level routinely keeps in touch as a necessity. it's quite possible that Sandman knew what went down shortly after it happened.