The white house's website, (and more particularly, a section dedicated to publishing official statements) is not a public forum.

The comment section of tweets is. And how government entities engage in public forums is regulated by the constitution and case law. Excluding people for previous expressions of their first amendment rights violates the first amendment in two rights, as it punishes the citizen for how they exercised their 1st amendment rights, in a way that further limits their 1st amendment rights.

This is akin to town halls ran by the government in private buildings. The building owner can surely exclude you because they don't like what you say, but the government can not.