Brazil beat Messi's Argentina in the semifinals and clinched the title Sunday with a 3-1 win over Peru for its first Copa since 2007 and ninth overall, ESPN reported.
"He has to have more respect and needs to accept when he loses," said Tite, who had praised the Argentine playmaker ahead of the semifinals.
Messi boycotted the medal ceremony after Argentina beat Chile in the third-place game on Saturday. He said he was protesting against the refereeing and "corruption" in the Copa America.
Messi and Chile's Gary Medel were red carded after their physical confrontation in the first half. Messi said afterward that Argentina was not treated fairly and hinted that Brazil was benefiting as a result.
Tite said he didn't think Messi should have been sent off against Chile, and that a yellow card would have been enough.
"You have to have a little more respect and be careful," Tite said.
"We played clean against Argentina. Messi was unfairly sent off against Chile. It was a yellow card. But be careful."
Brazil captain Dani Alves said that the comments by his former Barcelona teammate shouldn't take away from Brazil's win.
"I do not agree with what [Messi] said. Here we have professional players who have worked a lot in this competition and who have fought for it. Deservedly [Brazil] eliminated [Argentina in the semifinal]. This is a very special achievement. We do not have to think about what other people say, even if it's Messi."