I tried function adding console.log("now", this.state.likedPosts); line to check the state after updating at same time. That is why this problem arises.

  handleLike(movieId) {
    if (this.state.likedPosts.filter((e) => e.movie === movieId).length > 0) {
      const likedPosts = this.state.likedPosts.filter(function (l) {
        if (l.movie !== movieId) {
          return l;
        }
      });
      console.log(likedPosts); // this returns the expected object
      this.setState({ likedPosts }); // this not work
    } else {
      this.state.likedPosts.push({ movie: movieId, down: true }); // this works
    }
    console.log("now", this.state.likedPosts);
  }

Although the state was updated until the function was completed, the old state appeared to be there.

When this is done, the state is not updated in the same function, but when the state is compared using another function at same time, it seems to be working correctly.

You should never modify/mutate state directly (this.state.something = something). You should always use setState method.

To do something after setState you should use second argument of setState.

this.setState({
    likedPosts
}, () => {
    console.log(likedPosts)
});

Also, I would recommend you to start using react hooks.