Tony could be initially shocked. He might not understand why Peter would run away, especially if they had a good relationship before. He'd probably start searching for Peter right away, using all his resources.
One possible reason could be that Peter might feel like he's a burden to Tony. Maybe he made a big mistake and is afraid of Tony's reaction. For example, if he accidentally endangered the lives of others during a mission while under Tony's supervision.
Maybe Tony believes that Peter is in some kind of danger that May can't protect him from. So, he takes Peter away to keep him safe in a more secure and technologically advanced place that he can provide.
She may also be worried sick about Peter. She might try to find ways to get Peter back, like contacting other superhero friends of Peter or even going to the media if she has to.
In fanfiction, she might run away because of a misunderstanding. Maybe she overheard something she wasn't supposed to and misinterpreted it. For example, if she heard her father talking about some kind of protection plan for her that she thought was him trying to control her life more. It could also be that she has her own secrets, like developing powers that she's afraid to tell her father about and decides to run away to figure things out on her own.
He might first find a place to hide, like an abandoned building. There, he starts to realize how difficult it is to be on his own. He has to find food and water by himself.
He would be extremely worried. Tony Stark, despite his tough exterior, loves his daughter deeply. He'd probably start by using all his resources to find her. He has access to advanced technology, so he'd likely use his Stark Industries gadgets to track her down.
Perhaps Naruto feels that Kushina is being overprotective. He wants to prove his independence and go on his own adventures without her constant supervision, so he decides to run away.
He might miss out on some crucial training. Jiraiya has a lot of knowledge and skills to pass on, and by running away, Naruto could fall behind in his development as a ninja.
One possible reason could be that Naruto might be afraid of the intense training Jiraiya has in store for him. Jiraiya is known for his strict and grueling training methods, and Naruto, being young and sometimes impatient, might just want to escape that for a while.