What is the ruling on liking a page that sometimes publishes disbelief, and does the one who does so commit an act of disbelief if he denies what is in it with his heart and follows it for the sake of other posts?

It is obligatory to refute and denounce a page that propagates disbelief for anyone who comes across it, and denouncing it in one's heart is not sufficient if one is capable of refuting it in writing. Liking a page that propagates both disbelief and good is a form of praising a disbeliever. This is permissible in principle, unless the 'like' is an endorsement of their disbelief, or it leads to misleading Muslims. It is not permissible to 'like' such a page, even if the intention is to endorse the good it propagates, because doing so deceives other visitors and exposes oneself to trials.