I don't worry, at least not in the sense that if the C's lose it will be because of Tatum not being able to handle the playoffs.
One thing that people are concerned with about Tatum now is that he seems to be hitting the rookie wall. I'm actually less concerned about that in the playoffs. No back-to-backs, 1-2 days off between every game (more if they wrap up a series early), less travel, easier to be focused facing the same opponent 4-7 times in a row, etc.
Also with Tatum playing SF/PF, I don't see a lot of potential problem match ups where the C's can't afford a couple of rookie mistakes or hide him if necessary. Against Toronto or Indiana, I'd expect Brown and Smart to be tasked with handling DeRozen and Oladipo. Miami, Philly, Detroit, and Charlotte have nobody of concern that Tatum would have to match up against. Porter's been playing well since Wall went down, and could be a challenge for Tatum, but I don't see him as a guy who would kill the C's, definitely not losing sleep over what he might do. That leaves Milwaukee (Giannis/Middleton/Parker) and LeBron and the Cavs. We know LeBron can basically destroy whoever he wants, but with him being the only wing threat in Cleveland, Tatum doesn't have to be the one solely responsible for trying to stop him. And God help us all if Giannis decides he's going to take over. So I only see 2 teams out of 7 I'd be overly concerned about in the East, and the C's might not even match up with any of them. Miami/Indiana/Toronto/Houston is probably my ideal playoff run that I think is in the realm of possibility.
So I think Tatum will be okay for the C's in the playoffs. He's not going to sink the ship.