Just his general reactions while sleeping to what is happening in the dream world are pretty heartwarming.
Dream's Deep, the apparent deepest part of the dream world (and of Luigi's subconscious). When you get there, it looks like Luigi didn't make it there, but his thoughts float around you - things like "Let me help, bro!", "I can be strong!", "I fight ghosts!" and "Don't leave me, bro!". A Luigi even materializes with the words "Take me with you!", and expresses his worry that it could be dangerous and that Mario might not be able to return if he goes ahead.
When Mario and Luigi offend the Dream Stone's spirit, it transforms into a giant robot and begins attacking them. Both Dreamy Luigi and Luigi are terrified, but his love for his brother helps to transform Dreamy Luigi into a giant in order to fight like he thinks he should. The sheer amount of brotherly love in this game is practically tearjerker worthy.
This is cemented by the voices from Dream's Deep returning, saying things such as "Big Bro's in danger!" and "I gotta help Big Bro!" while an echoing cry for Mario sounds from Luigi as the Luiginoids band together. Also, in Dream's Deep, another of Luigi's thoughts say "Hey! Nobody mess with my bro!" Even though he's often Overshadowed by Awesome, Luigi clearly loves and respects his brother, and desperately wants to help him in any way he can.
Taken to eleven before the final giant battle, which is against Bowser himself. Since Luigi knows who (and what) he's up against, he has no fears about the battle. Instead of being worried about his brother, his subconscious says such inspiring quotes as "we'll defeat him together" and "with my bro!"
The thing that seals the deal for the heart-warmingness of the Giant Bowser scene is Real Luigi on the bottom screen. In the past four giant battles Luigi has cowered in fear at the sight of the massive adversary he's about to fight. In the battle with his worst enemy, he doesn't cower. He doesn't cry. He hardly even FLINCHES. The Bros. nod in understanding, and bottom screen Luigi pulls his cap down with swagger, ready for the battle. Also doubles as A crowning moment of awesomeness up to eleven
Awakening the Zeekeeper. This is an ancient deity of unrivaled supernatural power, and he is far, far from happy at having been woken. He threatens the Bros. with oblivion, flapping his wings hard and in turn, nearly blowing them away into the abyss. At this, the Giant Luigi trigger appears again...except it's not for Giant Luigi. Rather, the army of Luiginoids bands together around Mario, using themselves to brace and shield their brother from the onslaught. Even after all the Luiginoids are blown away, Dreamy Luigi summons his selves together to deliver the Zeekeeper the most vengeful, punishing, and amazing Goomba Stomp you have ever seen. It should be noted Mario was staring the thing down by himself, and said guardian told him to "abandon hope". Not a good idea when Luigi is still in earshot. He gave fair warning: nobody messes with his big bro.
The Hooski proposing to his lover in Wakeport. Especially with the Mario Bros. and all the NPCs cheering him on.
When Sorrow Fist starts bawling due to his conviction that he'll lose against the Mario Bros. due to being a weakling, Luigi, who has had similar feelings over being a Lesser Star, actually tries to comfort the guy. At least, until he realizes he is giving a pep talk to his opponent.
At the very end of the game, Bowser is retreating with Kamek and his trio of mooks, all the while swearing revenge on the Bros. He curses out Mario's name, then goes on to add "Green 'sta... Luigi!", finally remembering the man in green's name after ten years and showing he now considers him on the same level as Mario as an antagonist. And to a lesser extent he doesn't blame his minions for his defeat and admits they did better than usual.
The fact Bowser's recollection of Luigi's name is a detail that sticks all the way to Mario & Luigi: Paper Jam shows that Bowser now has an established begrudging respect for him as an opponent. Granted, it's respect by one of Luigi's greatest enemies, but it's respect all the same.