Valmiki Ramayana mentions that when Ma Sita was rescued Shri Ram reacted coldly and said some very harsh words. He told her that her abduction was a blot on him and his family’s name and he cleared it by rescuing her and now she was free to go anywhere. She could take refuge with anyone and anywhere but couldn't stay with him. These harsh words shocked Mata and she felt terrible grief. She told Laxmana to lit a pyre as she wanted to prove she was dedicated to her husband only. Laxmana looked at Shri Ram and when he signalled Laxmana then only he lit the pyre. Mata entered the fire and then gods came and gave testimony of her purity. Then Shri Ram revealed that he always knew his beloved wife was pure and was devoted to him only but this test was needed so that no blot stays on her name. He said he knew Ravana couldn't touch her because she was protected by her moral power. So, Shri Ram had immense confidence that Mata was pure and that's why he let her enter fire. In fact if he had even a little bit doubt on her he would never let her enter the fire. So this is a pure misconception that he doubted her, he repeated this again and again both in Yuddha and Uttar Kand that he never ever had any doubts on her.
In Ramopakhyan of Mahabharata, he was similarly cold to her and then she felt grief but she didn't enter fire in this one, in fact the gods came and gave testimony of her purity and that was enough.
In the author’s opinion back then people really feared devas and any oath taken in front of fire was considered true because unlike Kalyuga people back then truly believed Fire to be the purest thing in the universe and a medium to connect with Devas. So probably Mata took an oath in front of fire and didn't literally enter the fire but it's just an opinion. It's not written in the sources.