However, the First Amendment says that although every person can say what they wish, it discludes harmful and hateful pictures or texts.

is this right?

I doubt that the Supreme Court would agree with your interpretation, because what is harmful or hateful to one person may be a thing of beauty to another. However, if you are asking about the grammar, "disclude" is not a word. "Excepts" (makes an exception for, or does not include) is better.

Try this ...

Although the First Amendment says that all people can say what they wish, it excludes harmful and hateful pictures or texts.

the word discludes should be replaced by excludes

No, it's not right for a couple of reasons...

"every person" is singular, and then you have "they" (plural). So you could either do "every person can say what he or she wishes" OR you could write "although people can say what they wish"

As others have noted, "discludes" is not common usage. It's better to go with "does not include"

But even that isn't great. You should write, "it does not cover a person's right to produce harmful pictures or texts."

Although the First Amendment says that every person can say what they wish, it excludes harmful or hateful pictures & statements.

change discludes to excludes (discludes isn't a word) otherwise, I think it looks fine.