Is it true that every act of worship is accepted regardless of belief?

The statement that worship is accepted from a disbeliever as it is accepted from a believer is absolutely false. The disbeliever's worship is not accepted in the Hereafter, nor is he rewarded for it, because his deeds have been rendered void by disbelief. Allah Almighty says: (And We shall turn to whatever deeds they did [in this life] and make them as scattered dust.) [Al-Furqan 25:23]. However, he benefits from righteous deeds in this world, as Allah feeds him by virtue of them. As for the believer, his deeds are preserved for him in the Hereafter, and he is also rewarded for them in this world.

The supplication of a disbeliever may be answered, especially in times of dire need or oppression, as Allah Almighty says: (And when they embark on ships, they supplicate Allah, making their faith sincere to Him. But when He delivers them to the land, at once they associate others with Him.) [Al-Ankabut 29:65], and as the Prophet, peace and blessings be upon him, said: (Beware of the supplication of the oppressed, even if he is a disbeliever, for there is no barrier between it and Allah.)

Belief is not tied to social or mental status, and a deed may be rejected from its doer if it is not sincere for Allah and in accordance with the Sharia.